AlgebraAQA GCSE Mathematics (8300)

Notation, Vocabulary and Manipulation

Work through the notes, try the practice questions, then take the quiz. The report tells you exactly what to revise next. (2026)

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Master the topic

Worked examples4 Key formulae4 Definitions6 Common mistakes4

Step-by-step worked examples

01.Expanding and simplifying double brackets
Foundation
Question
Expand and simplify $(x + 4)(x - 3)$ .
Solution
1. 1
  Apply FOIL — multiply First, Outer, Inner, Last terms.
  $(x)(x) + (x)(-3) + (4)(x) + (4)(-3)$
2. 2
  Write out all four products.
  $x^2 - 3x + 4x - 12$
3. 3
  Collect the like terms $-3x$ and $+4x$ .
  $x^2 + x - 12$
Answer
$x^2 + x - 12$
02.Factorising a quadratic $ax^2 + bx + c$
Higher
Question
Factorise $3x^2 + 11x + 6$ .
Solution
1. 1
  Find $ac = 3 \times 6 = 18$ . Find two numbers that multiply to 18 and add to 11: these are 2 and 9.
2. 2
  Split the middle term using 2 and 9.
  $3x^2 + 2x + 9x + 6$
3. 3
  Factorise in pairs — take out a common factor from each pair.
  $x(3x + 2) + 3(3x + 2)$
4. 4
  $(3x + 2)$ is common to both groups. Write as a product.
  $(x + 3)(3x + 2)$
Answer
$(x + 3)(3x + 2)$
Examiner note
Check by expanding: $(x+3)(3x+2) = 3x^2 + 2x + 9x + 6 = 3x^2 + 11x + 6$ ✓
03.Rearranging when the subject appears twice
Higher
Question
Make $x$ the subject of $y = \dfrac{x + 3}{x - 1}$ .
Solution
1. 1
  Multiply both sides by $(x - 1)$ to clear the fraction.
  $y(x - 1) = x + 3$
2. 2
  Expand the left side.
  $yx - y = x + 3$
3. 3
  Collect all terms containing $x$ on the left.
  $yx - x = y + 3$
4. 4
  Factorise — $x$ is a common factor.
  $x(y - 1) = y + 3$
5. 5
  Divide both sides by $(y - 1)$ .
  $x = \frac{y + 3}{y - 1}$
Answer
$x = \dfrac{y + 3}{y - 1}$
04.Proving an algebraic identity
Higher
Question
Prove that $(x + 3)^2 - (x - 1)^2 \equiv 8(x + 1)$ .
Solution
1. 1
  Start with the left-hand side only. Expand $(x+3)^2$ using $(a+b)^2 = a^2+2ab+b^2$ .
  $(x+3)^2 = x^2 + 6x + 9$
2. 2
  Expand $(x-1)^2$ .
  $(x-1)^2 = x^2 - 2x + 1$
3. 3
  Subtract the second expansion from the first.
  $x^2 + 6x + 9 - (x^2 - 2x + 1) = 8x + 8$
4. 4
  Factorise $8x + 8$ to match the right-hand side.
  $8(x + 1) = \text{RHS} \checkmark$
Answer
Identity proved — LHS simplifies to $8(x + 1)$ for all values of $x$ .
Examiner note
Always end a proof with a clear conclusion statement to secure the final mark.

Key formulae

Difference of two squares
$a^2 - b^2 = (a + b)(a - b)$
- $a$
  — first expression
- $b$
  — second expression
When to use
When you have a squared term minus another squared term with no middle term.
Example
$9x^2 - 25 = (3x+5)(3x-5)$
Perfect square (sum)
$(a + b)^2 = a^2 + 2ab + b^2$
- $a$
  — first term
- $b$
  — second term
When to use
When expanding or recognising the square of a sum.
Example
$(x + 3)^2 = x^2 + 6x + 9$
Perfect square (difference)
$(a - b)^2 = a^2 - 2ab + b^2$
- $a$
  — first term
- $b$
  — second term
When to use
When expanding or recognising the square of a difference.
Example
$(x - 4)^2 = x^2 - 8x + 16$
Quadratic factorisation
$x^2 + bx + c = (x + p)(x + q) \text{ where } pq = c \text{ and } p + q = b$
- $b$
  — coefficient of x
- $c$
  — constant term
- $p, q$
  — two numbers that multiply to c and add to b
When to use
To factorise a monic quadratic (leading coefficient 1).
Example
$x^2 + 7x + 10 = (x+2)(x+5)$ since $2 \times 5=10$ and $2+5=7$

Practice with flashcards

Card 1 of 60 known · 0 to review

Key definitions and keywords

Expression: A mathematical phrase containing numbers, letters and operations but no equals sign. Examples: $3x^2 - 2x + 1$ , $ab + c$ .
Equation: A statement that two expressions are equal, true for specific values of the variable. Example: $2x + 1 = 7$ is true only when $x = 3$ .
Identity: An algebraic statement true for all values of the variable(s), denoted by $\equiv$ . Example: $3(x + 2) \equiv 3x + 6$ .
Subject (of a formula): The variable that appears on its own on one side of the formula. In $v = u + at$ , $v$ is the subject.
Coefficient: The numerical factor multiplying a term. In $3x^2$ the coefficient is $3$ ; in $-x$ it is $-1$ .
Factor: An expression that divides exactly into another. $3$ and $x$ are both factors of $3x^2$ ; $(x+2)$ is a factor of $x^2+5x+6$ .

Common mistakes and misconceptions

Mistake
Writing $(x + 3)^2 = x^2 + 9$
Why it happens
Students square each term separately instead of expanding as $(x+3)(x+3)$ .
How to avoid it
Always expand fully: $(x+3)^2 = x^2 + 6x + 9$ . Use the identity $(a+b)^2 = a^2 + 2ab + b^2$ — the middle term $2ab$ is never zero unless $a$ or $b$ is zero.
Mistake
Substituting $a = -2$ and writing $-2^2 = -4$
Why it happens
The negative sign is treated as part of the base without using brackets.
How to avoid it
Always bracket negative substitutions: $(-2)^2 = 4$ . The square applies to the whole number including its sign.
Mistake
Expanding $-(x - 3)$ as $-x - 3$
Why it happens
Only the first term inside is multiplied by the negative sign.
How to avoid it
$-(x - 3) = -x + 3$ . The negative multiplies every term inside the bracket, turning $-3$ into $+3$ .
Mistake
Factorising $4x^2 + 6x$ as $2(2x^2 + 3x)$ instead of $2x(2x + 3)$
Why it happens
Only the numerical common factor is taken out, missing the common factor of $x$ .
How to avoid it
Always take out the highest common factor. Both terms share $2x$ , giving $2x(2x + 3)$ . Check: $2x \times 2x = 4x^2$ ✓ and $2x \times 3 = 6x$ ✓.

AlgebraAQA GCSE Mathematics (8300)

Notation, Vocabulary and Manipulation

Work through the notes, try the practice questions, then take the quiz. The report tells you exactly what to revise next. (2026)

PreviousNext

Master the topic

Worked examples4 Key formulae4 Definitions6 Common mistakes4

Step-by-step worked examples

01.Expanding and simplifying double brackets
Foundation
Question
Expand and simplify $(x + 4)(x - 3)$ .
Solution
1. 1
  Apply FOIL — multiply First, Outer, Inner, Last terms.
  $(x)(x) + (x)(-3) + (4)(x) + (4)(-3)$
2. 2
  Write out all four products.
  $x^2 - 3x + 4x - 12$
3. 3
  Collect the like terms $-3x$ and $+4x$ .
  $x^2 + x - 12$
Answer
$x^2 + x - 12$
02.Factorising a quadratic $ax^2 + bx + c$
Higher
Question
Factorise $3x^2 + 11x + 6$ .
Solution
1. 1
  Find $ac = 3 \times 6 = 18$ . Find two numbers that multiply to 18 and add to 11: these are 2 and 9.
2. 2
  Split the middle term using 2 and 9.
  $3x^2 + 2x + 9x + 6$
3. 3
  Factorise in pairs — take out a common factor from each pair.
  $x(3x + 2) + 3(3x + 2)$
4. 4
  $(3x + 2)$ is common to both groups. Write as a product.
  $(x + 3)(3x + 2)$
Answer
$(x + 3)(3x + 2)$
Examiner note
Check by expanding: $(x+3)(3x+2) = 3x^2 + 2x + 9x + 6 = 3x^2 + 11x + 6$ ✓
03.Rearranging when the subject appears twice
Higher
Question
Make $x$ the subject of $y = \dfrac{x + 3}{x - 1}$ .
Solution
1. 1
  Multiply both sides by $(x - 1)$ to clear the fraction.
  $y(x - 1) = x + 3$
2. 2
  Expand the left side.
  $yx - y = x + 3$
3. 3
  Collect all terms containing $x$ on the left.
  $yx - x = y + 3$
4. 4
  Factorise — $x$ is a common factor.
  $x(y - 1) = y + 3$
5. 5
  Divide both sides by $(y - 1)$ .
  $x = \frac{y + 3}{y - 1}$
Answer
$x = \dfrac{y + 3}{y - 1}$
04.Proving an algebraic identity
Higher
Question
Prove that $(x + 3)^2 - (x - 1)^2 \equiv 8(x + 1)$ .
Solution
1. 1
  Start with the left-hand side only. Expand $(x+3)^2$ using $(a+b)^2 = a^2+2ab+b^2$ .
  $(x+3)^2 = x^2 + 6x + 9$
2. 2
  Expand $(x-1)^2$ .
  $(x-1)^2 = x^2 - 2x + 1$
3. 3
  Subtract the second expansion from the first.
  $x^2 + 6x + 9 - (x^2 - 2x + 1) = 8x + 8$
4. 4
  Factorise $8x + 8$ to match the right-hand side.
  $8(x + 1) = \text{RHS} \checkmark$
Answer
Identity proved — LHS simplifies to $8(x + 1)$ for all values of $x$ .
Examiner note
Always end a proof with a clear conclusion statement to secure the final mark.

Key formulae

Difference of two squares
$a^2 - b^2 = (a + b)(a - b)$
- $a$
  — first expression
- $b$
  — second expression
When to use
When you have a squared term minus another squared term with no middle term.
Example
$9x^2 - 25 = (3x+5)(3x-5)$
Perfect square (sum)
$(a + b)^2 = a^2 + 2ab + b^2$
- $a$
  — first term
- $b$
  — second term
When to use
When expanding or recognising the square of a sum.
Example
$(x + 3)^2 = x^2 + 6x + 9$
Perfect square (difference)
$(a - b)^2 = a^2 - 2ab + b^2$
- $a$
  — first term
- $b$
  — second term
When to use
When expanding or recognising the square of a difference.
Example
$(x - 4)^2 = x^2 - 8x + 16$
Quadratic factorisation
$x^2 + bx + c = (x + p)(x + q) \text{ where } pq = c \text{ and } p + q = b$
- $b$
  — coefficient of x
- $c$
  — constant term
- $p, q$
  — two numbers that multiply to c and add to b
When to use
To factorise a monic quadratic (leading coefficient 1).
Example
$x^2 + 7x + 10 = (x+2)(x+5)$ since $2 \times 5=10$ and $2+5=7$

Practice with flashcards

Card 1 of 60 known · 0 to review

Key definitions and keywords

Expression: A mathematical phrase containing numbers, letters and operations but no equals sign. Examples: $3x^2 - 2x + 1$ , $ab + c$ .
Equation: A statement that two expressions are equal, true for specific values of the variable. Example: $2x + 1 = 7$ is true only when $x = 3$ .
Identity: An algebraic statement true for all values of the variable(s), denoted by $\equiv$ . Example: $3(x + 2) \equiv 3x + 6$ .
Subject (of a formula): The variable that appears on its own on one side of the formula. In $v = u + at$ , $v$ is the subject.
Coefficient: The numerical factor multiplying a term. In $3x^2$ the coefficient is $3$ ; in $-x$ it is $-1$ .
Factor: An expression that divides exactly into another. $3$ and $x$ are both factors of $3x^2$ ; $(x+2)$ is a factor of $x^2+5x+6$ .

Common mistakes and misconceptions

Mistake
Writing $(x + 3)^2 = x^2 + 9$
Why it happens
Students square each term separately instead of expanding as $(x+3)(x+3)$ .
How to avoid it
Always expand fully: $(x+3)^2 = x^2 + 6x + 9$ . Use the identity $(a+b)^2 = a^2 + 2ab + b^2$ — the middle term $2ab$ is never zero unless $a$ or $b$ is zero.
Mistake
Substituting $a = -2$ and writing $-2^2 = -4$
Why it happens
The negative sign is treated as part of the base without using brackets.
How to avoid it
Always bracket negative substitutions: $(-2)^2 = 4$ . The square applies to the whole number including its sign.
Mistake
Expanding $-(x - 3)$ as $-x - 3$
Why it happens
Only the first term inside is multiplied by the negative sign.
How to avoid it
$-(x - 3) = -x + 3$ . The negative multiplies every term inside the bracket, turning $-3$ into $+3$ .
Mistake
Factorising $4x^2 + 6x$ as $2(2x^2 + 3x)$ instead of $2x(2x + 3)$
Why it happens
Only the numerical common factor is taken out, missing the common factor of $x$ .
How to avoid it
Always take out the highest common factor. Both terms share $2x$ , giving $2x(2x + 3)$ . Check: $2x \times 2x = 4x^2$ ✓ and $2x \times 3 = 6x$ ✓.

Notation, Vocabulary and Manipulation

Master the topic

Step-by-step worked examples

01.Expanding and simplifying double brackets

02.Factorising a quadratic ax2+bx+cax^2 + bx + cax2+bx+c

03.Rearranging when the subject appears twice

04.Proving an algebraic identity

Key formulae

Practice with flashcards

Key definitions and keywords

Common mistakes and misconceptions

Notation, Vocabulary and Manipulation

Master the topic

Step-by-step worked examples

01.Expanding and simplifying double brackets

02.Factorising a quadratic ax2+bx+cax^2 + bx + cax2+bx+c

03.Rearranging when the subject appears twice

04.Proving an algebraic identity

Key formulae

Practice with flashcards

Key definitions and keywords

Common mistakes and misconceptions

02.Factorising a quadratic $ax^2 + bx + c$

02.Factorising a quadratic $ax^2 + bx + c$