8 min readPathway sheet2-mark and 3-mark questions

Solving by factorisation

Split the middle term into two parts whose sum is b and product is ac. Factorise, then set each factor equal to zero.

Splitting ruleFind p and q with p + q = b and pq = ac

Zero product ruleIf (x − α)(x − β) = 0, then x = α or x = β

CheckPut each root back into the equation

Method: splitting the middle term

1
Write the equation in standard form ax² + bx + c = 0.
2
Find the product ac.
3★
Find two numbers p and q with p + q = b and pq = ac.
4
Write bx as px + qx.
5
Group the terms in pairs and take out the common factors.
6
Set each factor equal to zero and solve.

★ key step

Example 1: when a = 1

Problem

Solve x² − 3x − 28 = 0 by factorisation.

Working

ac = −28. Two numbers with product −28 and sum −3 are −7 and 4.
x² − 7x + 4x − 28 = 0
x(x − 7) + 4(x − 7) = 0
(x − 7)(x + 4) = 0
So x = 7 or x = −4.
Check: 49 − 21 − 28 = 0 and 16 + 12 − 28 = 0.

Answer

x = 7 or x = −4

Example 2: when a is not 1

Problem

Solve 6x² + x − 2 = 0 by factorisation.

Working

ac = 6 × (−2) = −12. Two numbers with product −12 and sum 1 are 4 and −3.
6x² + 4x − 3x − 2 = 0
2x(3x + 2) − 1(3x + 2) = 0
(3x + 2)(2x − 1) = 0
So x = −2/3 or x = 1/2.
Check x = 1/2: 6(1/4) + 1/2 − 2 = 3/2 + 1/2 − 2 = 0.

Answer

x = −2/3 or x = 1/2

Example 3: with square roots in the coefficients

Problem

Solve √5 x² + 6x + √5 = 0 by factorisation.

Working

ac = √5 × √5 = 5. Two numbers with product 5 and sum 6 are 5 and 1.
√5 x² + 5x + x + √5 = 0
Write 5x as √5 × √5 x: √5 x(x + √5) + 1(x + √5) = 0
(√5 x + 1)(x + √5) = 0
So x = −1/√5 = −√5/5, or x = −√5.
Check x = −√5: √5(5) + 6(−√5) + √5 = 5√5 − 6√5 + √5 = 0.

Answer

x = −√5/5 or x = −√5

Example 4: a word problem

Problem

The product of two consecutive positive even integers is 168. Find the integers.

Working

Let the integers be x and x + 2.
x(x + 2) = 168, so x² + 2x − 168 = 0.
Two numbers with product −168 and sum 2 are 14 and −12.
x² + 14x − 12x − 168 = x(x + 14) − 12(x + 14) = (x + 14)(x − 12) = 0
x = 12 or x = −14. Reject −14, since the integers are positive.
Check: 12 × 14 = 168.

Answer

The integers are 12 and 14.

In real life

Splitting the middle term means finding two numbers from their total and their product. It is how you find the two sides of a plot from its fencing (perimeter) and its area.

Exam angle

2 marks: “Solve by factorisation: 2x² − 7x + 3 = 0.” ac = 6; −6 and −1. 2x² − 6x − x + 3 = (2x − 1)(x − 3) = 0, so x = 1/2 or 3.

2 marks: “Find the roots of 4x² − 4x + 1 = 0.” 4x² − 2x − 2x + 1 = (2x − 1)(2x − 1) = 0, so x = 1/2, 1/2 (equal roots).

3 marks: “The sum of a number and its reciprocal is 26/5. Find the number.” x + 1/x = 26/5 gives 5x² − 26x + 5 = 0, so (5x − 1)(x − 5) = 0. The number is 5 or 1/5.

Remember it

Product ac, sum b: find the pair, split, group, and set each bracket to zero.