Expanding and Factorising Quadratics

Expanding Products of Two Binomials

Understanding how to multiply brackets explains why algebraic expressions can be reshaped to solve complex equations. A binomial is an algebraic expression consisting of two terms, such as $(x + 5)$ or $(2x - 3)$ .

To expand an expression means to remove the brackets by multiplying every term inside by the terms outside or in another bracket. When expanding the product (the result of multiplication) of two binomials, you will always perform 4 distinct multiplications before simplifying. You must then collect like terms, which are terms with the same variable raised to the same power, to form a quadratic trinomial (an expression with three terms).

There are two main methods for expanding two binomials:

1. The FOIL Method FOIL is a mnemonic to remember the four multiplications: First, Outside, Inside, Last.

Expand and simplify $(2x - 3)(x + 4)$

Step 1: Multiply the First terms.

$2x \times x = 2x^2$

Step 2: Multiply the Outside terms.

$2x \times 4 = +8x$

Step 3: Multiply the Inside terms.

$-3 \times x = -3x$

Step 4: Multiply the Last terms.

$-3 \times 4 = -12$

Step 5: Write out the full sum and collect like terms.

$2x^2 + 8x - 3x - 12$
$2x^2 + 5x - 12$

2. The Grid Method (Area Model) This method uses a 2x2 grid, placing one binomial across the top and the other down the side.

Expand and simplify $(x + 3)(x + 2)$

Step 1: Draw a grid and multiply the rows by the columns.

$\times$	$x$	$+2$
$x$	$x^2$	$+2x$
$+3$	$+3x$	$+6$

Step 2: Write out the terms from the grid and collect like terms (often found on the diagonal).

$x^2 + 2x + 3x + 6$
$x^2 + 5x + 6$

Expanding Products of Three Binomials (Higher Tier)

You can multiply two expressions easily, but try multiplying three without breaking it down into stages. Expanding three binomials is a Higher Tier requirement that relies on the distributive law, the rule used to multiply a single term or bracket by every term inside another bracket.

The process requires two stages. First, expand two binomials to create a quadratic. Then, systematically distribute this resulting quadratic into the third binomial, which requires 6 separate multiplications before final simplification.

Expand and simplify $(x + 2)(x - 3)(2x + 1)$

Step 1: Stage 1 — Expand and simplify the first two brackets.

$(x + 2)(x - 3) = x^2 - 3x + 2x - 6$
$= x^2 - x - 6$

Step 2: Stage 2 — Multiply the resulting quadratic by the third bracket.

$(x^2 - x - 6)(2x + 1)$

Step 3: Distribute every term systematically.

$x^2(2x) = 2x^3$
$x^2(1) = x^2$
$-x(2x) = -2x^2$
$-x(1) = -x$
$-6(2x) = -12x$
$-6(1) = -6$

Step 4: Collect all like terms to form your final polynomial (an expression consisting of variables and coefficients with non-negative powers).

$2x^3 + (x^2 - 2x^2) + (-x - 12x) - 6$
$2x^3 - x^2 - 13x - 6$

Factorising Unitary Quadratics

How do you reverse the process of expanding brackets to find the original factors? To factorise an expression means to rewrite it as a product of its factors (this process is known as factorisation).

A monic/unitary quadratic is a quadratic where the leading coefficient (the numerical value multiplied by the $x^2$ variable) is exactly 1, written in the form $x^2 + bx + c$ . To factorise these, use the Factor Pair Rule: you must find two integers ( $p$ and $q$ ) that multiply to give the constant term $c$ and add to give the coefficient $b$ .

Factorise $x^2 - 5x + 6$

Step 1: Identify $b$ and $c$ .

$b = -5$
$c = 6$

Step 2: Find factors of $c$ that add to $b$ . Because $c$ is positive and $b$ is negative, both factors must be negative.

Factors of $6$ : $(-1 \times -6)$ and $(-2 \times -3)$
Sum check: $-2 + -3 = -5$

Step 3: Write the factors in double brackets.

$(x - 2)(x - 3)$

Factorising Non-Unitary Quadratics

Every time you encounter a non-unitary quadratic (a quadratic where the leading number isn't 1), you need a highly systematic approach to break it apart. For expressions in the form $ax^2 + bx + c$ where $a \neq 1$ , use the AC Method / Decomposition.

This method involves splitting the middle term $bx$ into two separate terms, allowing you to use factorising by grouping — a technique where you split a four-term expression into two pairs and extract the highest common factor from each.

Factorise $6x^2 - 7x - 3$

Step 1: Identify $a$ , $b$ , and $c$ , then calculate $ac$ .

$a = 6$ , $b = -7$ , $c = -3$
$a \times c = 6 \times -3 = -18$

Step 2: Find two integers that multiply to $-18$ and add to $b$ ( $-7$ ).

The factors are $-9$ and $2$ .

Step 3: Decompose the middle term ( $-7x$ ) into $-9x$ and $+2x$ .

$6x^2 - 9x + 2x - 3$

Step 4: Group the four terms into two pairs and factorise each pair.

First pair: $3x(2x - 3)$
Second pair: $+1(2x - 3)$

Step 5: Factor out the common bracket.

$(3x + 1)(2x - 3)$

Factorising the Difference of Two Squares (DOTS)

The difference of two squares is one of the only algebraic rules that allows you to factorise an expression with just two terms. The Difference of Two Squares (DOTS) applies when a binomial consists of a perfect square (a term that is the square of another term) subtracted from another perfect square.

The rule to memorise is:

a^2 - b^2 = (a + b)(a - b)

Factorise fully $25y^2 - 9$

Step 1: Verify both terms are perfect squares separated by a subtraction sign.

$25y^2 = (5y)^2$
$9 = (3)^2$

Step 2: Apply the DOTS formula substituting $a = 5y$ and $b = 3$ .

$(5y + 3)(5y - 3)$

Factorise fully $2x^2 - 50$

Step 1: Look for a Highest Common Factor (HCF) first.

Factor out $2$ : $2(x^2 - 25)$

Step 2: Apply DOTS to the expression inside the bracket.

$x^2 = (x)^2$ and $25 = (5)^2$
$2(x + 5)(x - 5)$

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Exam Tips

Common Mistake
Students often incorrectly expand (x + 3)² as x² + 9. You must always rewrite it as (x + 3)(x + 3) and perform all four multiplications.
2
In Edexcel 'Show That' questions for expanding three binomials, examiners expect you to show every intermediate step of the distribution for full marks.
3
For method marks (M1) when expanding double brackets, you can often earn 1 mark for showing at least 3 out of 4 correct terms, even if your final simplification is incorrect.
4
When factorising non-unitary quadratics, quickly check your answer by expanding the final brackets to ensure they return to the original expression.
5
The phrase 'Factorise Fully' in an Edexcel exam is a major hint to check for a numerical Highest Common Factor (HCF) before applying the Difference of Two Squares.

Key Terms(18)

Binomial: An algebraic expression consisting of two terms, such as (x + 5) or (2x - 3).
Expand: To remove brackets by multiplying every term inside the bracket by the term(s) outside or in another bracket.
Product: The result of two or more terms or expressions being multiplied together.
Like terms: Terms with the same variable raised to the same power, which can be collected (added or subtracted) together.
Quadratic trinomial: A quadratic algebraic expression consisting of exactly three terms, typically in the form ax² + bx + c.
Polynomial: An algebraic expression consisting of variables and coefficients with non-negative integer powers.
Distributive law: The mathematical rule used to multiply a single term or bracket by every term inside another bracket.
Factorisation: The process of rewriting an expression as a product of its factors, which is the inverse of expanding.
Factorise: To perform factorisation; rewriting an expression as a product of its factors.
Coefficient: The numerical value that is multiplied by a variable in an algebraic term.
Constant term: A number in an algebraic expression that does not change its value because it contains no variables.
Monic/Unitary quadratic: A quadratic expression where the coefficient of the x² term is exactly 1.
Non-unitary quadratic: A quadratic expression where the coefficient of the x² term is not exactly 1.
Highest Common Factor: The largest factor that divides two or more algebraic terms or numbers exactly.
AC Method / Decomposition: A method for factorising non-unitary quadratics by splitting the middle term (bx) into two separate terms to allow for grouping.
Factorising by grouping: A factorisation technique that involves splitting a four-term expression into two pairs and extracting the highest common factor out of each pair.
Difference of Two Squares (DOTS): A factorisation pattern used for a binomial where one perfect square term is subtracted from another, following the rule a² - b² = (a + b)(a - b).
Perfect square: A number or algebraic term that is the exact square of another term, such as 9, 25y², or x⁴.

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Key Terms(18)

Binomial: An algebraic expression consisting of two terms, such as (x + 5) or (2x - 3).
Expand: To remove brackets by multiplying every term inside the bracket by the term(s) outside or in another bracket.
Product: The result of two or more terms or expressions being multiplied together.
Like terms: Terms with the same variable raised to the same power, which can be collected (added or subtracted) together.
Quadratic trinomial: A quadratic algebraic expression consisting of exactly three terms, typically in the form ax² + bx + c.
Polynomial: An algebraic expression consisting of variables and coefficients with non-negative integer powers.
Distributive law: The mathematical rule used to multiply a single term or bracket by every term inside another bracket.
Factorisation: The process of rewriting an expression as a product of its factors, which is the inverse of expanding.
Factorise: To perform factorisation; rewriting an expression as a product of its factors.
Coefficient: The numerical value that is multiplied by a variable in an algebraic term.
Constant term: A number in an algebraic expression that does not change its value because it contains no variables.
Monic/Unitary quadratic: A quadratic expression where the coefficient of the x² term is exactly 1.
Non-unitary quadratic: A quadratic expression where the coefficient of the x² term is not exactly 1.
Highest Common Factor: The largest factor that divides two or more algebraic terms or numbers exactly.
AC Method / Decomposition: A method for factorising non-unitary quadratics by splitting the middle term (bx) into two separate terms to allow for grouping.
Factorising by grouping: A factorisation technique that involves splitting a four-term expression into two pairs and extracting the highest common factor out of each pair.
Difference of Two Squares (DOTS): A factorisation pattern used for a binomial where one perfect square term is subtracted from another, following the rule a² - b² = (a + b)(a - b).
Perfect square: A number or algebraic term that is the exact square of another term, such as 9, 25y², or x⁴.

Expanding and Factorising Quadratics

Expanding Products of Two Binomials

There are two main methods for expanding two binomials:

1. The FOIL Method FOIL is a mnemonic to remember the four multiplications: First, Outside, Inside, Last.

Expand and simplify $(2x - 3)(x + 4)$

Step 1: Multiply the First terms.

$2x \times x = 2x^2$

Step 2: Multiply the Outside terms.

$2x \times 4 = +8x$

Step 3: Multiply the Inside terms.

$-3 \times x = -3x$

Step 4: Multiply the Last terms.

$-3 \times 4 = -12$

Step 5: Write out the full sum and collect like terms.

$2x^2 + 8x - 3x - 12$
$2x^2 + 5x - 12$

2. The Grid Method (Area Model) This method uses a 2x2 grid, placing one binomial across the top and the other down the side.

Expand and simplify $(x + 3)(x + 2)$

Step 1: Draw a grid and multiply the rows by the columns.

$\times$	$x$	$+2$
$x$	$x^2$	$+2x$
$+3$	$+3x$	$+6$

Step 2: Write out the terms from the grid and collect like terms (often found on the diagonal).

$x^2 + 2x + 3x + 6$
$x^2 + 5x + 6$

Expanding Products of Three Binomials (Higher Tier)

Expand and simplify $(x + 2)(x - 3)(2x + 1)$

Step 1: Stage 1 — Expand and simplify the first two brackets.

$(x + 2)(x - 3) = x^2 - 3x + 2x - 6$
$= x^2 - x - 6$

Step 2: Stage 2 — Multiply the resulting quadratic by the third bracket.

$(x^2 - x - 6)(2x + 1)$

Step 3: Distribute every term systematically.

$x^2(2x) = 2x^3$
$x^2(1) = x^2$
$-x(2x) = -2x^2$
$-x(1) = -x$
$-6(2x) = -12x$
$-6(1) = -6$

Step 4: Collect all like terms to form your final polynomial (an expression consisting of variables and coefficients with non-negative powers).

$2x^3 + (x^2 - 2x^2) + (-x - 12x) - 6$
$2x^3 - x^2 - 13x - 6$

Factorising Unitary Quadratics

Factorise $x^2 - 5x + 6$

Step 1: Identify $b$ and $c$ .

$b = -5$
$c = 6$

Step 2: Find factors of $c$ that add to $b$ . Because $c$ is positive and $b$ is negative, both factors must be negative.

Factors of $6$ : $(-1 \times -6)$ and $(-2 \times -3)$
Sum check: $-2 + -3 = -5$

Step 3: Write the factors in double brackets.

$(x - 2)(x - 3)$

Factorising Non-Unitary Quadratics

Factorise $6x^2 - 7x - 3$

Step 1: Identify $a$ , $b$ , and $c$ , then calculate $ac$ .

$a = 6$ , $b = -7$ , $c = -3$
$a \times c = 6 \times -3 = -18$

Step 2: Find two integers that multiply to $-18$ and add to $b$ ( $-7$ ).

The factors are $-9$ and $2$ .

Step 3: Decompose the middle term ( $-7x$ ) into $-9x$ and $+2x$ .

$6x^2 - 9x + 2x - 3$

Step 4: Group the four terms into two pairs and factorise each pair.

First pair: $3x(2x - 3)$
Second pair: $+1(2x - 3)$

Step 5: Factor out the common bracket.

$(3x + 1)(2x - 3)$

Factorising the Difference of Two Squares (DOTS)

The rule to memorise is:

a^2 - b^2 = (a + b)(a - b)

Factorise fully $25y^2 - 9$

Step 1: Verify both terms are perfect squares separated by a subtraction sign.

$25y^2 = (5y)^2$
$9 = (3)^2$

Step 2: Apply the DOTS formula substituting $a = 5y$ and $b = 3$ .

$(5y + 3)(5y - 3)$

Factorise fully $2x^2 - 50$

Step 1: Look for a Highest Common Factor (HCF) first.

Factor out $2$ : $2(x^2 - 25)$

Step 2: Apply DOTS to the expression inside the bracket.

$x^2 = (x)^2$ and $25 = (5)^2$
$2(x + 5)(x - 5)$

Sign up to continue reading

Get full access to revision notes, key terms, and exam tips for every subtopic.

Exam Tips

Common Mistake
Students often incorrectly expand (x + 3)² as x² + 9. You must always rewrite it as (x + 3)(x + 3) and perform all four multiplications.
2
In Edexcel 'Show That' questions for expanding three binomials, examiners expect you to show every intermediate step of the distribution for full marks.
3
For method marks (M1) when expanding double brackets, you can often earn 1 mark for showing at least 3 out of 4 correct terms, even if your final simplification is incorrect.
4
When factorising non-unitary quadratics, quickly check your answer by expanding the final brackets to ensure they return to the original expression.
5
The phrase 'Factorise Fully' in an Edexcel exam is a major hint to check for a numerical Highest Common Factor (HCF) before applying the Difference of Two Squares.

Key Terms(18)

Binomial: An algebraic expression consisting of two terms, such as (x + 5) or (2x - 3).
Expand: To remove brackets by multiplying every term inside the bracket by the term(s) outside or in another bracket.
Product: The result of two or more terms or expressions being multiplied together.
Like terms: Terms with the same variable raised to the same power, which can be collected (added or subtracted) together.
Quadratic trinomial: A quadratic algebraic expression consisting of exactly three terms, typically in the form ax² + bx + c.
Polynomial: An algebraic expression consisting of variables and coefficients with non-negative integer powers.
Distributive law: The mathematical rule used to multiply a single term or bracket by every term inside another bracket.
Factorisation: The process of rewriting an expression as a product of its factors, which is the inverse of expanding.
Factorise: To perform factorisation; rewriting an expression as a product of its factors.
Coefficient: The numerical value that is multiplied by a variable in an algebraic term.
Constant term: A number in an algebraic expression that does not change its value because it contains no variables.
Monic/Unitary quadratic: A quadratic expression where the coefficient of the x² term is exactly 1.
Non-unitary quadratic: A quadratic expression where the coefficient of the x² term is not exactly 1.
Highest Common Factor: The largest factor that divides two or more algebraic terms or numbers exactly.
AC Method / Decomposition: A method for factorising non-unitary quadratics by splitting the middle term (bx) into two separate terms to allow for grouping.
Factorising by grouping: A factorisation technique that involves splitting a four-term expression into two pairs and extracting the highest common factor out of each pair.
Difference of Two Squares (DOTS): A factorisation pattern used for a binomial where one perfect square term is subtracted from another, following the rule a² - b² = (a + b)(a - b).
Perfect square: A number or algebraic term that is the exact square of another term, such as 9, 25y², or x⁴.

Previous NoteBasic Algebraic Manipulation and Indices Next NoteSurds and Algebraic Fractions

Back to Expression Simplification

Put your knowledge into practice — try past paper questions for Mathematics

Key Terms(18)

Binomial: An algebraic expression consisting of two terms, such as (x + 5) or (2x - 3).
Expand: To remove brackets by multiplying every term inside the bracket by the term(s) outside or in another bracket.
Product: The result of two or more terms or expressions being multiplied together.
Like terms: Terms with the same variable raised to the same power, which can be collected (added or subtracted) together.
Quadratic trinomial: A quadratic algebraic expression consisting of exactly three terms, typically in the form ax² + bx + c.
Polynomial: An algebraic expression consisting of variables and coefficients with non-negative integer powers.
Distributive law: The mathematical rule used to multiply a single term or bracket by every term inside another bracket.
Factorisation: The process of rewriting an expression as a product of its factors, which is the inverse of expanding.
Factorise: To perform factorisation; rewriting an expression as a product of its factors.
Coefficient: The numerical value that is multiplied by a variable in an algebraic term.
Constant term: A number in an algebraic expression that does not change its value because it contains no variables.
Monic/Unitary quadratic: A quadratic expression where the coefficient of the x² term is exactly 1.
Non-unitary quadratic: A quadratic expression where the coefficient of the x² term is not exactly 1.
Highest Common Factor: The largest factor that divides two or more algebraic terms or numbers exactly.
AC Method / Decomposition: A method for factorising non-unitary quadratics by splitting the middle term (bx) into two separate terms to allow for grouping.
Factorising by grouping: A factorisation technique that involves splitting a four-term expression into two pairs and extracting the highest common factor out of each pair.
Difference of Two Squares (DOTS): A factorisation pattern used for a binomial where one perfect square term is subtracted from another, following the rule a² - b² = (a + b)(a - b).
Perfect square: A number or algebraic term that is the exact square of another term, such as 9, 25y², or x⁴.