Higher only · completing the square

Complete the
square in
three moves.

halve it, square it,
take it away

the form you are aiming for —

(x + p)² + q

a squared bracket

plus or minus a number

step 1 of 4

Halve the x number

1. Halve the number in front of x.

2. That half goes inside the bracket.

✓ (x + 3)² is the start

x² + 6x + 2

6 ÷ 2 = 3

(x + 3)²

step 2 of 4

Take away the square

1. The bracket makes an extra + 9.

2. Subtract 9, then bring down the + 2.

✓ −9 + 2 = −7

(x + 3)² = x² + 6x + 9

(x + 3)² − 9 + 2

= (x + 3)² − 7

step 3 of 4

Solve with it

1. Move the number across.

2. Square root both sides. Remember ±.

✓ Exact answers, left as surds

(x + 3)² − 7 = 0

(x + 3)² = 7

x + 3 = ±√7

x = −3 ± √7

step 4 of 4

Read the turning point

1. A square is smallest when it is 0.

2. So x = −3, and y is what is left: −7.

✓ Flip the sign inside the bracket

y = (x + 3)² − 7

x = −3 makes the bracket 0

turning point (−3, −7)

completing the square —

Halve

the x number, into the bracket

Subtract

that half squared

(x + p)² + q

turning point (−p, q)

Odd number? Halve it anyway.
Fractions are fine.

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