Completing the square

The Maths Habit
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Sam says (x + 3)2 = x2 + 9. Is Sam right?

No: (x + 3)2 = x2 + 6x + 9

Expand

  1. (x + 5)2

    x2 + 10x + 25

  2. (x − 4)2

    x2 − 8x + 16

  3. (x + 1)2

    x2 + 2x + 1

x2 + 12x as areas

x²xx1212x

A square x by x and a strip x by 12

Halve the 12x: 6x on each of two sides

x²6x6xx6x636

The square has a corner missing. Missing: 6 × 6 = 36

Fill the corner, then take it away again

x²6x6xx6x636x + 6

x2 + 12x = (x + 6)2 − 36

Quick way: halve the number of x, take away its square.

Completing the square

  1. HalveHalve the number of x.
  2. The bracketPut the half in a bracket, squared.
  3. Take away the squareSquare the half and take it away.
    Keep the + c.
  4. AnswerWork out the numbers at the end.
x2 + bx + c
half = b ÷ 2
(x + half)2
(x + half)2 − half2 + c
(x + half)2 + one number
x² − bx? the half is negative
(x + half)² gives x² + bx + half², one number too many
take away half², add c, as one number
Walkthrough to print

Complete the square for x2 + 4x + 1.

halve the number of x2
(x +2)2 −4 + 1
Answer(x + 2)2 − 3

Complete the square for x2 + 10x + 7.

halve the number of x5
(x +5)2 −25 + 7
Answer(x + 5)2 − 18

Complete the square for x2 + 4x + 1.

halve the number of x2
(x +2)2 −4 + 1
Answer(x + 2)2 − 3

Complete the square for x2 − 8x + 3.

halve the number of x−4
(x −4)2 −16 + 3
Answer(x − 4)2 − 13

What is the turning point?

y = (x + 3)2 − 4 (−3, −4)

  1. Write down the turning point of y = (x − 5)2 + 2.

    (5, 2)

  2. Write down the turning point of y = (x + 1)2 − 9.

    (−1, −9)

  3. What is the smallest value of (x − 4)2 + 11?

    11, when x = 4

  4. Complete the square to find the turning point of y = x2 + 8x + 20.

    (−4, 4)

Completing the square worksheet, page 1

Answer sheet

  1. Solve x2 + 6x + 2 = 0 by completing the square. Leave your answer in surd form.

    −3 ± √7

  2. What is the smallest value of x2 − 10x + 30?

    5

  1. Write x2 + 8x + 3 in the form (x + p)2 + q

    (x + 4)2 − 13

  2. Write down the turning point of y = (x + 6)2 − 2

    (−6, −2)

  3. Complete the square to find the turning point of y = x2 − 10x + 18

    (5, −7)
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