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Mathematics · Quadratic Equations

Flip through 20 full sample pages with substantial teaching notes, deep-dive explanations, worked practice, exam application and self-check questions.

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Overview

Algebra, functions and problem solving

This sample is designed to show how strong mathematics notes should work: concise theory, exact notation, worked reasoning, common mistakes and practice. The focus is algebraic fluency because it underpins equations, graphs, calculus, statistics and many applied problems.

  • Simplify before substituting.
  • Keep equality balanced when solving.
  • Use exact values where possible.
  • Check answers by substitution or estimation.
Worked example / practice: Warm-up: solve 3x + 5 = 20. Subtract 5, divide by 3, so x = 5.

Deep-dive notes

The central idea on this page is algebra, functions and problem solving. To use it confidently, connect the definition or rule above to the specific details listed here: Simplify before substituting.; Keep equality balanced when solving.; Use exact values where possible.. These are not separate facts to memorise in isolation; they form the reasoning chain you should be able to reconstruct without looking.

A strong revision method is to close the notes after reading this section and reproduce the key idea from memory. Then compare your version with the page, identify any missing terminology, and correct the explanation before moving to practice. This converts passive reading into active retrieval and makes the page useful for both first learning and later revision.

Exam-style application

Possible question: Explain, apply or use the idea of “Algebra, functions and problem solving” in a new situation. Start by stating the relevant rule or definition precisely, then use the information in the question, show the intermediate reasoning, and finish with a conclusion that answers the command word.

  1. Identify exactly what the question is asking and underline the command word.
  2. Write the relevant definition, relationship, rule or principle before substituting or applying it.
  3. Use the information given rather than relying on vague general statements.
  4. Show the reasoning in a logical sequence so method marks remain visible.
  5. Check the final answer for units, terminology, plausibility and relevance to the question.

Before you turn the page

  • Can you define or explain algebra, functions and problem solving without looking?
  • Can you give one correct example and one common mistake?
  • Can you recognise when this idea should be used in a question?
  • Can you explain your method clearly enough for someone else to follow?
Exam / study tip: Mathematics marks are often lost through skipped algebra rather than difficult concepts. Write one logical transformation per line.
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Foundations

Algebraic notation and terms

An expression contains numbers, variables and operations but no equality sign. An equation states that two expressions are equal. A term is a part separated by addition or subtraction; coefficients multiply variables and constants contain no variable.

  • In 5x² − 3x + 7, the terms are 5x², −3x and 7.
  • 5 is the coefficient of x².
  • Like terms have identical variable parts.
  • Only like terms may be collected directly.
Worked example / practice: Example: 4x + 3 − 2x + 5 = 2x + 8.

Deep-dive notes

The central idea on this page is algebraic notation and terms. To use it confidently, connect the definition or rule above to the specific details listed here: In 5x² − 3x + 7, the terms are 5x², −3x and 7.; 5 is the coefficient of x².; Like terms have identical variable parts.. These are not separate facts to memorise in isolation; they form the reasoning chain you should be able to reconstruct without looking.

A strong revision method is to close the notes after reading this section and reproduce the key idea from memory. Then compare your version with the page, identify any missing terminology, and correct the explanation before moving to practice. This converts passive reading into active retrieval and makes the page useful for both first learning and later revision.

Exam-style application

Possible question: Explain, apply or use the idea of “Algebraic notation and terms” in a new situation. Start by stating the relevant rule or definition precisely, then use the information in the question, show the intermediate reasoning, and finish with a conclusion that answers the command word.

  1. Identify exactly what the question is asking and underline the command word.
  2. Write the relevant definition, relationship, rule or principle before substituting or applying it.
  3. Use the information given rather than relying on vague general statements.
  4. Show the reasoning in a logical sequence so method marks remain visible.
  5. Check the final answer for units, terminology, plausibility and relevance to the question.

Before you turn the page

  • Can you define or explain algebraic notation and terms without looking?
  • Can you give one correct example and one common mistake?
  • Can you recognise when this idea should be used in a question?
  • Can you explain your method clearly enough for someone else to follow?
Exam / study tip: Do not combine x and x². They are unlike terms.
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Expanding

Single and double brackets

Expansion uses the distributive law. Every term outside a bracket multiplies every term inside it. For two binomials, systematic multiplication prevents missing terms.

  • a(b+c)=ab+ac.
  • (x+3)(x+5)=x²+8x+15.
  • Track negative signs carefully.
  • Collect like terms after expansion.
Worked example / practice: Worked: (2x−3)(x+4)=2x²+8x−3x−12=2x²+5x−12.

Deep-dive notes

The central idea on this page is single and double brackets. To use it confidently, connect the definition or rule above to the specific details listed here: a(b+c)=ab+ac.; (x+3)(x+5)=x²+8x+15.; Track negative signs carefully.. These are not separate facts to memorise in isolation; they form the reasoning chain you should be able to reconstruct without looking.

A strong revision method is to close the notes after reading this section and reproduce the key idea from memory. Then compare your version with the page, identify any missing terminology, and correct the explanation before moving to practice. This converts passive reading into active retrieval and makes the page useful for both first learning and later revision.

Exam-style application

Possible question: Explain, apply or use the idea of “Single and double brackets” in a new situation. Start by stating the relevant rule or definition precisely, then use the information in the question, show the intermediate reasoning, and finish with a conclusion that answers the command word.

  1. Identify exactly what the question is asking and underline the command word.
  2. Write the relevant definition, relationship, rule or principle before substituting or applying it.
  3. Use the information given rather than relying on vague general statements.
  4. Show the reasoning in a logical sequence so method marks remain visible.
  5. Check the final answer for units, terminology, plausibility and relevance to the question.

Before you turn the page

  • Can you define or explain single and double brackets without looking?
  • Can you give one correct example and one common mistake?
  • Can you recognise when this idea should be used in a question?
  • Can you explain your method clearly enough for someone else to follow?
Exam / study tip: Write the four products before collecting. Mental expansion is faster only when it is also accurate.
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Factorising

Factorising as reverse expansion

Factorising rewrites an expression as a product. Start by looking for a highest common factor. For quadratics x²+bx+c, find two numbers whose product is c and sum is b.

  • 6x²+9x = 3x(2x+3).
  • x²+7x+12=(x+3)(x+4).
  • Difference of squares: a²−b²=(a−b)(a+b).
  • Always expand your factorisation mentally to check.
Worked example / practice: Example: 2x²+8x = 2x(x+4).

Deep-dive notes

The central idea on this page is factorising as reverse expansion. To use it confidently, connect the definition or rule above to the specific details listed here: 6x²+9x = 3x(2x+3).; x²+7x+12=(x+3)(x+4).; Difference of squares: a²−b²=(a−b)(a+b).. These are not separate facts to memorise in isolation; they form the reasoning chain you should be able to reconstruct without looking.

A strong revision method is to close the notes after reading this section and reproduce the key idea from memory. Then compare your version with the page, identify any missing terminology, and correct the explanation before moving to practice. This converts passive reading into active retrieval and makes the page useful for both first learning and later revision.

Exam-style application

Possible question: Explain, apply or use the idea of “Factorising as reverse expansion” in a new situation. Start by stating the relevant rule or definition precisely, then use the information in the question, show the intermediate reasoning, and finish with a conclusion that answers the command word.

  1. Identify exactly what the question is asking and underline the command word.
  2. Write the relevant definition, relationship, rule or principle before substituting or applying it.
  3. Use the information given rather than relying on vague general statements.
  4. Show the reasoning in a logical sequence so method marks remain visible.
  5. Check the final answer for units, terminology, plausibility and relevance to the question.

Before you turn the page

  • Can you define or explain factorising as reverse expansion without looking?
  • Can you give one correct example and one common mistake?
  • Can you recognise when this idea should be used in a question?
  • Can you explain your method clearly enough for someone else to follow?
Exam / study tip: Never cancel terms across addition. Factor first if cancellation is needed.
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Linear equations

Solving linear equations

Solving an equation means finding values that make it true. Whatever operation is applied to one side must be applied to the other side. Simplify each side before isolating the variable.

  • Remove brackets first.
  • Collect variable terms on one side.
  • Collect constants on the other.
  • Divide by the coefficient last.
Worked example / practice: Worked: 3(2x−1)=15 → 6x−3=15 → 6x=18 → x=3.

Deep-dive notes

The central idea on this page is solving linear equations. To use it confidently, connect the definition or rule above to the specific details listed here: Remove brackets first.; Collect variable terms on one side.; Collect constants on the other.. These are not separate facts to memorise in isolation; they form the reasoning chain you should be able to reconstruct without looking.

A strong revision method is to close the notes after reading this section and reproduce the key idea from memory. Then compare your version with the page, identify any missing terminology, and correct the explanation before moving to practice. This converts passive reading into active retrieval and makes the page useful for both first learning and later revision.

Exam-style application

Possible question: Explain, apply or use the idea of “Solving linear equations” in a new situation. Start by stating the relevant rule or definition precisely, then use the information in the question, show the intermediate reasoning, and finish with a conclusion that answers the command word.

  1. Identify exactly what the question is asking and underline the command word.
  2. Write the relevant definition, relationship, rule or principle before substituting or applying it.
  3. Use the information given rather than relying on vague general statements.
  4. Show the reasoning in a logical sequence so method marks remain visible.
  5. Check the final answer for units, terminology, plausibility and relevance to the question.

Before you turn the page

  • Can you define or explain solving linear equations without looking?
  • Can you give one correct example and one common mistake?
  • Can you recognise when this idea should be used in a question?
  • Can you explain your method clearly enough for someone else to follow?
Exam / study tip: Check by substituting x=3 into the original equation, not only the final simplified line.
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Fractions

Algebraic fractions

Treat algebraic fractions like numerical fractions while respecting restrictions. Factor expressions before cancellation and use a common denominator when adding or subtracting.

  • Cancel factors, not terms.
  • State excluded values where relevant.
  • For addition, build a common denominator.
  • Simplify the final result.
Worked example / practice: Example: (x²−9)/(x−3) = (x−3)(x+3)/(x−3)=x+3, provided x≠3.

Deep-dive notes

The central idea on this page is algebraic fractions. To use it confidently, connect the definition or rule above to the specific details listed here: Cancel factors, not terms.; State excluded values where relevant.; For addition, build a common denominator.. These are not separate facts to memorise in isolation; they form the reasoning chain you should be able to reconstruct without looking.

A strong revision method is to close the notes after reading this section and reproduce the key idea from memory. Then compare your version with the page, identify any missing terminology, and correct the explanation before moving to practice. This converts passive reading into active retrieval and makes the page useful for both first learning and later revision.

Exam-style application

Possible question: Explain, apply or use the idea of “Algebraic fractions” in a new situation. Start by stating the relevant rule or definition precisely, then use the information in the question, show the intermediate reasoning, and finish with a conclusion that answers the command word.

  1. Identify exactly what the question is asking and underline the command word.
  2. Write the relevant definition, relationship, rule or principle before substituting or applying it.
  3. Use the information given rather than relying on vague general statements.
  4. Show the reasoning in a logical sequence so method marks remain visible.
  5. Check the final answer for units, terminology, plausibility and relevance to the question.

Before you turn the page

  • Can you define or explain algebraic fractions without looking?
  • Can you give one correct example and one common mistake?
  • Can you recognise when this idea should be used in a question?
  • Can you explain your method clearly enough for someone else to follow?
Exam / study tip: The restriction remains even after cancellation because the original expression was undefined at x=3.
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Simultaneous equations

Two equations, two unknowns

Simultaneous equations seek values satisfying both equations. Use elimination when coefficients align conveniently and substitution when one variable is already isolated.

  • Elimination: add/subtract equations to remove one variable.
  • Substitution: replace one variable using an equivalent expression.
  • Solve for the remaining variable.
  • Substitute back and check both equations.
Worked example / practice: Worked: x+y=7 and x−y=1. Add: 2x=8, so x=4; then y=3.

Deep-dive notes

The central idea on this page is two equations, two unknowns. To use it confidently, connect the definition or rule above to the specific details listed here: Elimination: add/subtract equations to remove one variable.; Substitution: replace one variable using an equivalent expression.; Solve for the remaining variable.. These are not separate facts to memorise in isolation; they form the reasoning chain you should be able to reconstruct without looking.

A strong revision method is to close the notes after reading this section and reproduce the key idea from memory. Then compare your version with the page, identify any missing terminology, and correct the explanation before moving to practice. This converts passive reading into active retrieval and makes the page useful for both first learning and later revision.

Exam-style application

Possible question: Explain, apply or use the idea of “Two equations, two unknowns” in a new situation. Start by stating the relevant rule or definition precisely, then use the information in the question, show the intermediate reasoning, and finish with a conclusion that answers the command word.

  1. Identify exactly what the question is asking and underline the command word.
  2. Write the relevant definition, relationship, rule or principle before substituting or applying it.
  3. Use the information given rather than relying on vague general statements.
  4. Show the reasoning in a logical sequence so method marks remain visible.
  5. Check the final answer for units, terminology, plausibility and relevance to the question.

Before you turn the page

  • Can you define or explain two equations, two unknowns without looking?
  • Can you give one correct example and one common mistake?
  • Can you recognise when this idea should be used in a question?
  • Can you explain your method clearly enough for someone else to follow?
Exam / study tip: A solution pair must satisfy both equations.
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Quadratics

Solving quadratic equations

A quadratic equation has highest power 2. Bring all terms to one side before factorising, completing the square or using the quadratic formula.

  • Standard form: ax²+bx+c=0.
  • Factorisation works when convenient factors exist.
  • Quadratic formula works generally.
  • A quadratic may have two, one or no real roots.
Worked example / practice: Example: x²−5x+6=0 → (x−2)(x−3)=0, so x=2 or x=3.

Deep-dive notes

The central idea on this page is solving quadratic equations. To use it confidently, connect the definition or rule above to the specific details listed here: Standard form: ax²+bx+c=0.; Factorisation works when convenient factors exist.; Quadratic formula works generally.. These are not separate facts to memorise in isolation; they form the reasoning chain you should be able to reconstruct without looking.

A strong revision method is to close the notes after reading this section and reproduce the key idea from memory. Then compare your version with the page, identify any missing terminology, and correct the explanation before moving to practice. This converts passive reading into active retrieval and makes the page useful for both first learning and later revision.

Exam-style application

Possible question: Explain, apply or use the idea of “Solving quadratic equations” in a new situation. Start by stating the relevant rule or definition precisely, then use the information in the question, show the intermediate reasoning, and finish with a conclusion that answers the command word.

  1. Identify exactly what the question is asking and underline the command word.
  2. Write the relevant definition, relationship, rule or principle before substituting or applying it.
  3. Use the information given rather than relying on vague general statements.
  4. Show the reasoning in a logical sequence so method marks remain visible.
  5. Check the final answer for units, terminology, plausibility and relevance to the question.

Before you turn the page

  • Can you define or explain solving quadratic equations without looking?
  • Can you give one correct example and one common mistake?
  • Can you recognise when this idea should be used in a question?
  • Can you explain your method clearly enough for someone else to follow?
Exam / study tip: Never stop at the factorised form if the question asks you to solve.
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Formula

The quadratic formula

For ax²+bx+c=0, x=[−b±√(b²−4ac)]/(2a). The discriminant b²−4ac indicates the nature of the roots.

  • Discriminant >0: two distinct real roots.
  • =0: one repeated real root.
  • <0: no real roots in the real-number system.
  • Use brackets around negative b values when substituting.
Worked example / practice: Example: x²+2x−3=0 gives x=[−2±√16]/2, so x=1 or −3.

Deep-dive notes

The central idea on this page is the quadratic formula. To use it confidently, connect the definition or rule above to the specific details listed here: Discriminant >0: two distinct real roots.; =0: one repeated real root.; <0: no real roots in the real-number system.. These are not separate facts to memorise in isolation; they form the reasoning chain you should be able to reconstruct without looking.

A strong revision method is to close the notes after reading this section and reproduce the key idea from memory. Then compare your version with the page, identify any missing terminology, and correct the explanation before moving to practice. This converts passive reading into active retrieval and makes the page useful for both first learning and later revision.

Exam-style application

Possible question: Explain, apply or use the idea of “The quadratic formula” in a new situation. Start by stating the relevant rule or definition precisely, then use the information in the question, show the intermediate reasoning, and finish with a conclusion that answers the command word.

  1. Identify exactly what the question is asking and underline the command word.
  2. Write the relevant definition, relationship, rule or principle before substituting or applying it.
  3. Use the information given rather than relying on vague general statements.
  4. Show the reasoning in a logical sequence so method marks remain visible.
  5. Check the final answer for units, terminology, plausibility and relevance to the question.

Before you turn the page

  • Can you define or explain the quadratic formula without looking?
  • Can you give one correct example and one common mistake?
  • Can you recognise when this idea should be used in a question?
  • Can you explain your method clearly enough for someone else to follow?
Exam / study tip: Keep the ± until the final two branches are evaluated.
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Functions

Function notation

A function assigns each input in its domain exactly one output. Function notation f(x) describes the rule and allows evaluation, equations and transformations to be expressed compactly.

  • f(3) means substitute x=3.
  • f(a+h) means substitute a+h everywhere x appears.
  • Solve f(x)=k by setting the expression equal to k.
  • Domain restrictions matter.
Worked example / practice: If f(x)=2x²−1, then f(3)=17 and f(a)=2a²−1.

Deep-dive notes

The central idea on this page is function notation. To use it confidently, connect the definition or rule above to the specific details listed here: f(3) means substitute x=3.; f(a+h) means substitute a+h everywhere x appears.; Solve f(x)=k by setting the expression equal to k.. These are not separate facts to memorise in isolation; they form the reasoning chain you should be able to reconstruct without looking.

A strong revision method is to close the notes after reading this section and reproduce the key idea from memory. Then compare your version with the page, identify any missing terminology, and correct the explanation before moving to practice. This converts passive reading into active retrieval and makes the page useful for both first learning and later revision.

Exam-style application

Possible question: Explain, apply or use the idea of “Function notation” in a new situation. Start by stating the relevant rule or definition precisely, then use the information in the question, show the intermediate reasoning, and finish with a conclusion that answers the command word.

  1. Identify exactly what the question is asking and underline the command word.
  2. Write the relevant definition, relationship, rule or principle before substituting or applying it.
  3. Use the information given rather than relying on vague general statements.
  4. Show the reasoning in a logical sequence so method marks remain visible.
  5. Check the final answer for units, terminology, plausibility and relevance to the question.

Before you turn the page

  • Can you define or explain function notation without looking?
  • Can you give one correct example and one common mistake?
  • Can you recognise when this idea should be used in a question?
  • Can you explain your method clearly enough for someone else to follow?
Exam / study tip: Do not interpret f(x) as f multiplied by x.
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Graphs

Straight-line graphs

The equation y=mx+c describes a straight line, where m is gradient and c is the y-intercept. Gradient is change in y divided by change in x.

  • Positive m: line rises left to right.
  • Negative m: line falls left to right.
  • Parallel lines have equal gradients.
  • Perpendicular non-vertical lines have gradients whose product is −1.
Worked example / practice: Through (2,5) and (6,13): m=(13−5)/(6−2)=2. Then 5=2(2)+c, so c=1; y=2x+1.

Deep-dive notes

The central idea on this page is straight-line graphs. To use it confidently, connect the definition or rule above to the specific details listed here: Positive m: line rises left to right.; Negative m: line falls left to right.; Parallel lines have equal gradients.. These are not separate facts to memorise in isolation; they form the reasoning chain you should be able to reconstruct without looking.

A strong revision method is to close the notes after reading this section and reproduce the key idea from memory. Then compare your version with the page, identify any missing terminology, and correct the explanation before moving to practice. This converts passive reading into active retrieval and makes the page useful for both first learning and later revision.

Exam-style application

Possible question: Explain, apply or use the idea of “Straight-line graphs” in a new situation. Start by stating the relevant rule or definition precisely, then use the information in the question, show the intermediate reasoning, and finish with a conclusion that answers the command word.

  1. Identify exactly what the question is asking and underline the command word.
  2. Write the relevant definition, relationship, rule or principle before substituting or applying it.
  3. Use the information given rather than relying on vague general statements.
  4. Show the reasoning in a logical sequence so method marks remain visible.
  5. Check the final answer for units, terminology, plausibility and relevance to the question.

Before you turn the page

  • Can you define or explain straight-line graphs without looking?
  • Can you give one correct example and one common mistake?
  • Can you recognise when this idea should be used in a question?
  • Can you explain your method clearly enough for someone else to follow?
Exam / study tip: Use widely separated points on a drawn best-fit line when estimating gradient.
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Transformations

Graph transformations

Transformations change the position or scale of a graph. The direction can feel counter-intuitive inside the function argument, so learn the mapping rather than guessing.

  • y=f(x)+a shifts up by a.
  • y=f(x−a) shifts right by a.
  • y=af(x) stretches vertically by factor a.
  • y=f(ax) scales horizontally by factor 1/a.
Worked example / practice: Example: y=(x−3)² is y=x² shifted 3 units right.

Deep-dive notes

The central idea on this page is graph transformations. To use it confidently, connect the definition or rule above to the specific details listed here: y=f(x)+a shifts up by a.; y=f(x−a) shifts right by a.; y=af(x) stretches vertically by factor a.. These are not separate facts to memorise in isolation; they form the reasoning chain you should be able to reconstruct without looking.

A strong revision method is to close the notes after reading this section and reproduce the key idea from memory. Then compare your version with the page, identify any missing terminology, and correct the explanation before moving to practice. This converts passive reading into active retrieval and makes the page useful for both first learning and later revision.

Exam-style application

Possible question: Explain, apply or use the idea of “Graph transformations” in a new situation. Start by stating the relevant rule or definition precisely, then use the information in the question, show the intermediate reasoning, and finish with a conclusion that answers the command word.

  1. Identify exactly what the question is asking and underline the command word.
  2. Write the relevant definition, relationship, rule or principle before substituting or applying it.
  3. Use the information given rather than relying on vague general statements.
  4. Show the reasoning in a logical sequence so method marks remain visible.
  5. Check the final answer for units, terminology, plausibility and relevance to the question.

Before you turn the page

  • Can you define or explain graph transformations without looking?
  • Can you give one correct example and one common mistake?
  • Can you recognise when this idea should be used in a question?
  • Can you explain your method clearly enough for someone else to follow?
Exam / study tip: Inside changes act on x and therefore appear in the opposite horizontal direction.
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Indices

Laws of indices

Index laws compress repeated multiplication. They apply to the same base and are essential for algebra, scientific notation and exponentials.

  • a^m a^n = a^(m+n).
  • a^m/a^n=a^(m−n), a≠0.
  • (a^m)^n=a^(mn).
  • a^0=1 and a^(−n)=1/a^n.
Worked example / practice: Example: x³·x⁵/x²=x^(3+5−2)=x⁶.

Deep-dive notes

The central idea on this page is laws of indices. To use it confidently, connect the definition or rule above to the specific details listed here: a^m a^n = a^(m+n).; a^m/a^n=a^(m−n), a≠0.; (a^m)^n=a^(mn).. These are not separate facts to memorise in isolation; they form the reasoning chain you should be able to reconstruct without looking.

A strong revision method is to close the notes after reading this section and reproduce the key idea from memory. Then compare your version with the page, identify any missing terminology, and correct the explanation before moving to practice. This converts passive reading into active retrieval and makes the page useful for both first learning and later revision.

Exam-style application

Possible question: Explain, apply or use the idea of “Laws of indices” in a new situation. Start by stating the relevant rule or definition precisely, then use the information in the question, show the intermediate reasoning, and finish with a conclusion that answers the command word.

  1. Identify exactly what the question is asking and underline the command word.
  2. Write the relevant definition, relationship, rule or principle before substituting or applying it.
  3. Use the information given rather than relying on vague general statements.
  4. Show the reasoning in a logical sequence so method marks remain visible.
  5. Check the final answer for units, terminology, plausibility and relevance to the question.

Before you turn the page

  • Can you define or explain laws of indices without looking?
  • Can you give one correct example and one common mistake?
  • Can you recognise when this idea should be used in a question?
  • Can you explain your method clearly enough for someone else to follow?
Exam / study tip: Do not add powers when adding terms: x²+x²=2x², not x⁴.
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Surds

Exact values and surds

A surd is an irrational root kept in exact radical form. Simplify by extracting perfect-square factors and rationalise denominators when required.

  • √50=5√2.
  • √a√b=√(ab) for suitable non-negative values.
  • Like surds can be collected.
  • Rationalisation removes a surd from the denominator.
Worked example / practice: Example: 3√8−√18 = 6√2−3√2=3√2.

Deep-dive notes

The central idea on this page is exact values and surds. To use it confidently, connect the definition or rule above to the specific details listed here: √50=5√2.; √a√b=√(ab) for suitable non-negative values.; Like surds can be collected.. These are not separate facts to memorise in isolation; they form the reasoning chain you should be able to reconstruct without looking.

A strong revision method is to close the notes after reading this section and reproduce the key idea from memory. Then compare your version with the page, identify any missing terminology, and correct the explanation before moving to practice. This converts passive reading into active retrieval and makes the page useful for both first learning and later revision.

Exam-style application

Possible question: Explain, apply or use the idea of “Exact values and surds” in a new situation. Start by stating the relevant rule or definition precisely, then use the information in the question, show the intermediate reasoning, and finish with a conclusion that answers the command word.

  1. Identify exactly what the question is asking and underline the command word.
  2. Write the relevant definition, relationship, rule or principle before substituting or applying it.
  3. Use the information given rather than relying on vague general statements.
  4. Show the reasoning in a logical sequence so method marks remain visible.
  5. Check the final answer for units, terminology, plausibility and relevance to the question.

Before you turn the page

  • Can you define or explain exact values and surds without looking?
  • Can you give one correct example and one common mistake?
  • Can you recognise when this idea should be used in a question?
  • Can you explain your method clearly enough for someone else to follow?
Exam / study tip: Keep exact form unless the question specifically requests a decimal.
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Inequalities

Solving inequalities

Inequalities are manipulated like equations except that multiplying or dividing by a negative number reverses the inequality sign.

  • x+3>7 gives x>4.
  • −2x≤6 gives x≥−3.
  • Represent solutions on a number line when requested.
  • For quadratic inequalities, use critical points and sign regions.
Worked example / practice: Example: 3−x<7 → −x<4 → x>−4 after dividing by −1.

Deep-dive notes

The central idea on this page is solving inequalities. To use it confidently, connect the definition or rule above to the specific details listed here: x+3>7 gives x>4.; −2x≤6 gives x≥−3.; Represent solutions on a number line when requested.. These are not separate facts to memorise in isolation; they form the reasoning chain you should be able to reconstruct without looking.

A strong revision method is to close the notes after reading this section and reproduce the key idea from memory. Then compare your version with the page, identify any missing terminology, and correct the explanation before moving to practice. This converts passive reading into active retrieval and makes the page useful for both first learning and later revision.

Exam-style application

Possible question: Explain, apply or use the idea of “Solving inequalities” in a new situation. Start by stating the relevant rule or definition precisely, then use the information in the question, show the intermediate reasoning, and finish with a conclusion that answers the command word.

  1. Identify exactly what the question is asking and underline the command word.
  2. Write the relevant definition, relationship, rule or principle before substituting or applying it.
  3. Use the information given rather than relying on vague general statements.
  4. Show the reasoning in a logical sequence so method marks remain visible.
  5. Check the final answer for units, terminology, plausibility and relevance to the question.

Before you turn the page

  • Can you define or explain solving inequalities without looking?
  • Can you give one correct example and one common mistake?
  • Can you recognise when this idea should be used in a question?
  • Can you explain your method clearly enough for someone else to follow?
Exam / study tip: The reversed sign is a frequent exam trap.
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Sequences

Arithmetic and geometric sequences

Sequences are ordered lists generated by rules. Arithmetic sequences have a constant difference; geometric sequences have a constant ratio.

  • Arithmetic nth term: a+(n−1)d.
  • Geometric nth term: ar^(n−1).
  • Check the first few terms after deriving a formula.
  • Recurrence relations define terms from previous terms.
Worked example / practice: Arithmetic example 5,8,11,... has a=5,d=3, so u_n=5+3(n−1)=3n+2.

Deep-dive notes

The central idea on this page is arithmetic and geometric sequences. To use it confidently, connect the definition or rule above to the specific details listed here: Arithmetic nth term: a+(n−1)d.; Geometric nth term: ar^(n−1).; Check the first few terms after deriving a formula.. These are not separate facts to memorise in isolation; they form the reasoning chain you should be able to reconstruct without looking.

A strong revision method is to close the notes after reading this section and reproduce the key idea from memory. Then compare your version with the page, identify any missing terminology, and correct the explanation before moving to practice. This converts passive reading into active retrieval and makes the page useful for both first learning and later revision.

Exam-style application

Possible question: Explain, apply or use the idea of “Arithmetic and geometric sequences” in a new situation. Start by stating the relevant rule or definition precisely, then use the information in the question, show the intermediate reasoning, and finish with a conclusion that answers the command word.

  1. Identify exactly what the question is asking and underline the command word.
  2. Write the relevant definition, relationship, rule or principle before substituting or applying it.
  3. Use the information given rather than relying on vague general statements.
  4. Show the reasoning in a logical sequence so method marks remain visible.
  5. Check the final answer for units, terminology, plausibility and relevance to the question.

Before you turn the page

  • Can you define or explain arithmetic and geometric sequences without looking?
  • Can you give one correct example and one common mistake?
  • Can you recognise when this idea should be used in a question?
  • Can you explain your method clearly enough for someone else to follow?
Exam / study tip: Distinguish 'term number' n from the actual term value.
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Coordinate geometry

Distance, midpoint and lines

Coordinate geometry combines algebra with geometry. Use exact coordinates to calculate gradient, midpoint and distance, then connect these to line equations and geometric properties.

  • Midpoint=((x₁+x₂)/2,(y₁+y₂)/2).
  • Distance=√[(x₂−x₁)²+(y₂−y₁)²].
  • Gradient=(y₂−y₁)/(x₂−x₁).
  • Use gradients to prove parallel or perpendicular relationships.
Worked example / practice: For A(1,2), B(5,10): midpoint=(3,6), gradient=2, distance=√80=4√5.

Deep-dive notes

The central idea on this page is distance, midpoint and lines. To use it confidently, connect the definition or rule above to the specific details listed here: Midpoint=((x₁+x₂)/2,(y₁+y₂)/2).; Distance=√[(x₂−x₁)²+(y₂−y₁)²].; Gradient=(y₂−y₁)/(x₂−x₁).. These are not separate facts to memorise in isolation; they form the reasoning chain you should be able to reconstruct without looking.

A strong revision method is to close the notes after reading this section and reproduce the key idea from memory. Then compare your version with the page, identify any missing terminology, and correct the explanation before moving to practice. This converts passive reading into active retrieval and makes the page useful for both first learning and later revision.

Exam-style application

Possible question: Explain, apply or use the idea of “Distance, midpoint and lines” in a new situation. Start by stating the relevant rule or definition precisely, then use the information in the question, show the intermediate reasoning, and finish with a conclusion that answers the command word.

  1. Identify exactly what the question is asking and underline the command word.
  2. Write the relevant definition, relationship, rule or principle before substituting or applying it.
  3. Use the information given rather than relying on vague general statements.
  4. Show the reasoning in a logical sequence so method marks remain visible.
  5. Check the final answer for units, terminology, plausibility and relevance to the question.

Before you turn the page

  • Can you define or explain distance, midpoint and lines without looking?
  • Can you give one correct example and one common mistake?
  • Can you recognise when this idea should be used in a question?
  • Can you explain your method clearly enough for someone else to follow?
Exam / study tip: Label coordinates clearly before substitution to reduce sign errors.
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Problem solving

Turning words into mathematics

Multi-step problems test modelling more than arithmetic. Define unknowns, translate relationships into equations, solve, then interpret the solution in context and reject impossible values.

  • Define the variable in words.
  • Build an equation from the relationships.
  • Solve with full working.
  • Check units, range and contextual meaning.
Worked example / practice: Example: a rectangle has length x+3 and width x, area 40. x(x+3)=40 → x²+3x−40=0 → (x+8)(x−5)=0. Width must be positive, so x=5.

Deep-dive notes

The central idea on this page is turning words into mathematics. To use it confidently, connect the definition or rule above to the specific details listed here: Define the variable in words.; Build an equation from the relationships.; Solve with full working.. These are not separate facts to memorise in isolation; they form the reasoning chain you should be able to reconstruct without looking.

A strong revision method is to close the notes after reading this section and reproduce the key idea from memory. Then compare your version with the page, identify any missing terminology, and correct the explanation before moving to practice. This converts passive reading into active retrieval and makes the page useful for both first learning and later revision.

Exam-style application

Possible question: Explain, apply or use the idea of “Turning words into mathematics” in a new situation. Start by stating the relevant rule or definition precisely, then use the information in the question, show the intermediate reasoning, and finish with a conclusion that answers the command word.

  1. Identify exactly what the question is asking and underline the command word.
  2. Write the relevant definition, relationship, rule or principle before substituting or applying it.
  3. Use the information given rather than relying on vague general statements.
  4. Show the reasoning in a logical sequence so method marks remain visible.
  5. Check the final answer for units, terminology, plausibility and relevance to the question.

Before you turn the page

  • Can you define or explain turning words into mathematics without looking?
  • Can you give one correct example and one common mistake?
  • Can you recognise when this idea should be used in a question?
  • Can you explain your method clearly enough for someone else to follow?
Exam / study tip: Always return to the context after solving; algebra can produce mathematically valid but physically impossible roots.
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Exam technique

Showing mathematical reasoning

Method marks reward a correct process even if the final answer is wrong. Structure work so that each transformation can be followed.

  • Copy the equation accurately.
  • Write the formula before substitution.
  • Keep exact values until the last step.
  • Use ≈ only when rounding.
Worked example / practice: A calculator display of 3.141592... should not automatically become the answer; follow the requested significant figures or decimal places.

Deep-dive notes

The central idea on this page is showing mathematical reasoning. To use it confidently, connect the definition or rule above to the specific details listed here: Copy the equation accurately.; Write the formula before substitution.; Keep exact values until the last step.. These are not separate facts to memorise in isolation; they form the reasoning chain you should be able to reconstruct without looking.

A strong revision method is to close the notes after reading this section and reproduce the key idea from memory. Then compare your version with the page, identify any missing terminology, and correct the explanation before moving to practice. This converts passive reading into active retrieval and makes the page useful for both first learning and later revision.

Exam-style application

Possible question: Explain, apply or use the idea of “Showing mathematical reasoning” in a new situation. Start by stating the relevant rule or definition precisely, then use the information in the question, show the intermediate reasoning, and finish with a conclusion that answers the command word.

  1. Identify exactly what the question is asking and underline the command word.
  2. Write the relevant definition, relationship, rule or principle before substituting or applying it.
  3. Use the information given rather than relying on vague general statements.
  4. Show the reasoning in a logical sequence so method marks remain visible.
  5. Check the final answer for units, terminology, plausibility and relevance to the question.

Before you turn the page

  • Can you define or explain showing mathematical reasoning without looking?
  • Can you give one correct example and one common mistake?
  • Can you recognise when this idea should be used in a question?
  • Can you explain your method clearly enough for someone else to follow?
Exam / study tip: Box or underline the final answer and include units where relevant.
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Practice

Mixed algebra practice

Attempt these from a blank page: 1) Expand (3x−2)(x+5). 2) Solve 5x−7=18. 3) Factor x²−9x+20. 4) If f(x)=x²+2x, find f(−3). 5) Solve −3x>12.

  • 1) Multiply all four products.
  • 2) Isolate x.
  • 3) Product 20, sum −9.
  • 4) Substitute −3 with brackets.
  • 5) Reverse the sign after dividing by −3.
Worked example / practice: Answers: 1) 3x²+13x−10. 2) x=5. 3) (x−4)(x−5). 4) 3. 5) x<−4.

Deep-dive notes

The central idea on this page is mixed algebra practice. To use it confidently, connect the definition or rule above to the specific details listed here: 1) Multiply all four products.; 2) Isolate x.; 3) Product 20, sum −9.. These are not separate facts to memorise in isolation; they form the reasoning chain you should be able to reconstruct without looking.

A strong revision method is to close the notes after reading this section and reproduce the key idea from memory. Then compare your version with the page, identify any missing terminology, and correct the explanation before moving to practice. This converts passive reading into active retrieval and makes the page useful for both first learning and later revision.

Exam-style application

Possible question: Explain, apply or use the idea of “Mixed algebra practice” in a new situation. Start by stating the relevant rule or definition precisely, then use the information in the question, show the intermediate reasoning, and finish with a conclusion that answers the command word.

  1. Identify exactly what the question is asking and underline the command word.
  2. Write the relevant definition, relationship, rule or principle before substituting or applying it.
  3. Use the information given rather than relying on vague general statements.
  4. Show the reasoning in a logical sequence so method marks remain visible.
  5. Check the final answer for units, terminology, plausibility and relevance to the question.

Before you turn the page

  • Can you define or explain mixed algebra practice without looking?
  • Can you give one correct example and one common mistake?
  • Can you recognise when this idea should be used in a question?
  • Can you explain your method clearly enough for someone else to follow?
Exam / study tip: Redo any incorrect question without looking at the worked answer, then compare line by line.
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