Factorising Quadratics

Here is a place you can practise factorising quadratic equations assuming that you have already learnt to do so.

Let's begin with a reminder. Here's a revision video:

Video

Factorising an expression with a constant factor.

Example :: 40h + 88

Identify a common factor of 8 to give \(8(5h+11)\)

Level 1

Factorising an expression with a variable factor.

Example :: 23h + 3h²

Identify a common factor of h to give \(h(23+3h)\)

Level 2

Factorising an expression with both a constant and variable factor.

Example :: 10a + 2ab

Identify a common factor of 2a to give \(2a(5+b)\)

Level 3

Completely factorising an expression of two parts that can be separately factorised.

Example :: 3a + ab + 5w + 4wy

Identify common factors of pairs of terms to give \(a(3+b)+w(5+4y)\)

Level 4

Writing an expression as the product of two binomials.

Example :: 5a + 5b + ac + bc

Extract common factors of pairs of terms to give \(5(a+b) + c(a+b)\)
Then extract a common factor of (a+b) to give \((a+b)(5+c)\)

Level 5

Factorising an expression which is the difference between two squares.

Example :: c² - 81

This can be written as \( (c+9)(c-9) \). Expand the brackets to see why

Level 6

Factorising a quadratic where the squared term coefficient is 1.

Example :: c² - 13c + 36

Start by finding two numbers with a product of 36 and sum of -13
These numbers are -4 and -9
So the factorisation is \( (c-4)(c-9) \)

Level 7

Factorising a quadratic where the squared term coefficient is not 1.

Example :: 4a² + 5a - 6

Multiply the coefficient of a² by the constant term: \( 4 \times -6 = -24 \)
Now find two numbers with a product of -24 and sum of 5 (the coefficient of a)
These numbers are -3 and 8
Write the original expression with the middle term split into these two numbers: \( 4a^2-3a+8a-6\)
Extract common factors of pairs of terms to give \( a(4a-3) + 2(4a-3)\)
Then extract a common factor of (4a-3) to give \((a+2)(4a-3)\)

Level 8

Mixed factorisation questions

Example :: b² + 6b + 9

Use the techniques used in the previous levels

Level 9

However if you are a teacher would like access to a projectable set of questions to use for whole-class revision click the blue button below:

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Factorising quadratic expressions simplifies them by spliting them into simpler parts (factors). It is the reverse of expanding brackets. It is useful in the study of algebraic expressions and graphs and for solving quadratic equations.

There are many more algebra activities on Transum Maths including a more structured exercise on Factorising.

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