Foundation and Higher · solving quadratics

Solve a quadratic
by factorising.
Two answers.

each bracket
equals zero

the one fact that makes it work —

Zero

if two things multiply to make 0

one of them must be 0

step 1 of 4

Factorise first

1. Two numbers: multiply to 6, add to 5.

2. Set each bracket equal to 0.

✓ The sign flips: + 2 gives −2

x² + 5x + 6 = 0

(x + 2)(x + 3) = 0 × 6, + 5

x + 2 = 0  or  x + 3 = 0

x = −2  or  x = −3

step 2 of 4

With negatives

1. Multiply to −12: one is negative.

2. −4 and 3 add to −1. That is the pair.

✓ Check: 4² − 4 − 12 = 0

x² − x − 12 = 0

(x − 4)(x + 3) = 0 × −12, + −1

x − 4 = 0  or  x + 3 = 0

x = 4  or  x = −3

step 3 of 4

Make it equal 0 first

1. It only works when one side is 0.

2. Subtract 7x, add 10. Then factorise.

✓ Always rearrange to = 0 first

x² = 7x − 10

x² − 7x + 10 = 0 all on one side

(x − 2)(x − 5) = 0

x = 2  or  x = 5

step 4 of 4

Only two terms

1. No number term: take x out as a factor.

2. Do not divide by x: you lose an answer.

✓ x = 0 is a solution too

x² − 6x = 0

x(x − 6) = 0 take out x

x = 0  or  x = 6

solving by factorising —

= 0 first

everything on one side

Factorise

multiply to c, add to b

Each bracket = 0

two answers

The sign flips:
(x + 2) gives x = −2.

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