Sec 2 Maths · Expansion and factorisation of quadratic expressions

Special products and identities questions with answers

Practice questions on special products and identities, from the basics up to exam level, each with a full worked solution. Try them on paper first, then open the solution to check your working.

Worked example

Expand and simplify $(2x - 3)(x^{2} + x - 6)$ .

Step 1 of 2

Multiply each term of the first bracket by every term of the second

(2x - 3)(x^{2} + x - 6) = 2x^{3} + 2x^{2} - 12x - 3x^{2} - 3x + 18

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Questions

Question 1A bracket times a trinomial
Expand and simplify $(4x - 5)(3x^{2} + 6x + 1)$ .
Show the worked solution
1. Step 1: Multiply each term of the first bracket by every term of the second $(4x - 5)(3x^{2} + 6x + 1) = 12x^{3} + 24x^{2} + 4x - 15x^{2} - 30x - 5$
2. Step 2: Collect like terms $= 12x^{3} + 9x^{2} - 26x - 5$
Question 2Products in two letters
Expand and simplify $(5m - 4n)(3m + 4n)$ .
Show the worked solution
1. Step 1: Multiply each term of the first bracket by each term of the second $(5m - 4n)(3m + 4n) = 15m^{2} + 20mn - 12mn - 16n^{2}$
2. Step 2: Collect like terms $= 15m^{2} + 8mn - 16n^{2}$
Question 3Values from the identities
Given that $x + 4y = 24$ and $x - 4y = -32$ , find the value of $x^{2} - 16y^{2}$ .
Show the worked solution
1. Step 1: Write x² - 16y² using what you are given $x^{2} - 16y^{2} = (x + 4y)(x - 4y)$
2. Step 2: Substitute the values given $= (24)(-32)$
3. Step 3: Work it out $= -768$
Question 4a² − b² with numbers
Without using a calculator, evaluate $85^{2} - 15^{2}$ .
Show the worked solution
1. Step 1: Use a² − b² = (a + b)(a − b) $85^{2} - 15^{2} = (85 + 15)(85 - 15)$
2. Step 2: Work out the brackets, then multiply $= 100 \times 70$ $= 7000$
Question 5Show it is a multiple of k
Show that $(n + 1)^{2} - (n - 1)^{2}$ is a multiple of $4$ for every positive integer $n$ .
Show the worked solution
1. Step 1: Expand each square $(n + 1)^{2} - (n - 1)^{2} = (n^{2} + 2n + 1) - (n^{2} - 2n + 1)$
2. Step 2: Remove the brackets (the minus changes every sign in the second) and collect like terms $= 4n$ $= 4 \times n$ : $4$ times a whole number, so it is a multiple of $4$ for every positive integer $n$ .
Question 6Why it is always even, odd or a multiple
$n$ is a positive integer. Which statement explains why $n + (n + 1) + (n + 2)$ is always a multiple of 3?
Show the worked solution
1. Step 1: Simplify $n + (n + 1) + (n + 2) = 3(n + 1)$
2. Step 2: What does that form tell you?It simplifies to $3(n + 1)$ , which is $3$ times a whole number.

More on expansion and factorisation of quadratic expressions

Notes: Expansion and factorisation of quadratic expressions Worksheet with answers (PDF)

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Worked example

Expand and simplify

(2x - 3)(x^{2} + x - 6)

Step 1 of 2

Multiply each term of the first bracket by every term of the second

(2x - 3)(x^{2} + x - 6) = 2x^{3} + 2x^{2} - 12x - 3x^{2} - 3x + 18

Questions

Question 1A bracket times a trinomial

Expand and simplify $(4x - 5)(3x^{2} + 6x + 1)$ .

Show the worked solution

Step 1: Multiply each term of the first bracket by every term of the second $(4x - 5)(3x^{2} + 6x + 1) = 12x^{3} + 24x^{2} + 4x - 15x^{2} - 30x - 5$
Step 2: Collect like terms $= 12x^{3} + 9x^{2} - 26x - 5$

Question 2Products in two letters

Expand and simplify $(5m - 4n)(3m + 4n)$ .

Show the worked solution

Step 1: Multiply each term of the first bracket by each term of the second $(5m - 4n)(3m + 4n) = 15m^{2} + 20mn - 12mn - 16n^{2}$
Step 2: Collect like terms $= 15m^{2} + 8mn - 16n^{2}$

Question 3Values from the identities

Given that $x + 4y = 24$ and $x - 4y = -32$ , find the value of $x^{2} - 16y^{2}$ .

Show the worked solution

Step 1: Write x² - 16y² using what you are given $x^{2} - 16y^{2} = (x + 4y)(x - 4y)$
Step 2: Substitute the values given $= (24)(-32)$
Step 3: Work it out $= -768$

Question 4a² − b² with numbers

Without using a calculator, evaluate $85^{2} - 15^{2}$ .

Show the worked solution

Step 1: Use a² − b² = (a + b)(a − b) $85^{2} - 15^{2} = (85 + 15)(85 - 15)$
Step 2: Work out the brackets, then multiply $= 100 \times 70$ $= 7000$

Question 5Show it is a multiple of k

Show that $(n + 1)^{2} - (n - 1)^{2}$ is a multiple of $4$ for every positive integer $n$ .

Show the worked solution

Step 1: Expand each square $(n + 1)^{2} - (n - 1)^{2} = (n^{2} + 2n + 1) - (n^{2} - 2n + 1)$
Step 2: Remove the brackets (the minus changes every sign in the second) and collect like terms $= 4n$ $= 4 \times n$ : $4$ times a whole number, so it is a multiple of $4$ for every positive integer $n$ .

Question 6Why it is always even, odd or a multiple

$n$ is a positive integer. Which statement explains why $n + (n + 1) + (n + 2)$ is always a multiple of 3?

Show the worked solution

Step 1: Simplify $n + (n + 1) + (n + 2) = 3(n + 1)$
Step 2: What does that form tell you?It simplifies to $3(n + 1)$ , which is $3$ times a whole number.