Completing the square

Warm-up

Three quick questions from earlier lessons.

Discover

$x^2 + 6x + 9 = (x + 3)^2$ . Where does the 9 come from?

Not to scale

Step 1x² + 6x: a square and a strip.
Step 2Split the strip into two 3x strips and put them on two sides.
Step 3The corner is missing: add (6 ÷ 2)² = 9 to get (x + 3)².

$(6 \div 2)^2$
$6 + 3$

Key idea

Complete the square.

x^2 + bx + \left(\tfrac{b}{2}\right)^2 = \left(x + \tfrac{b}{2}\right)^2

. To solve

x^2 + bx = k

: add

\left(\tfrac{b}{2}\right)^2

to both sides, write the left side as a square, then take

\pm

square roots. The same move turns

x^2 + bx + c

into vertex form.

Worked example

Solve $x^2 + 4x = 1$ .

x^2 + 4x + 4 = 5

Add (4/2)² = 4 to both sides.

(x + 2)^2 = 5

Square.

x = -2 \pm \sqrt{5}

±√5, subtract 2.

Your turn, guided

Your turn: the number that completes $x^2 + 10x + \;?$ .

25

(10/2)².

Practice ★

What number makes $x^2 + 10x + \;?$ a perfect square?
What number makes $x^2 + 18x + \;?$ a perfect square?

$x^2 + 12x + 22 = (x + 6)^2 + \;?$

Watch out!

Practice ★★

Solve $x^2 + 6x = 8$ by completing the square. Enter all solutions (use sqrt).
Solve $x^2 + 2x = 9$ by completing the square. Enter all solutions (use sqrt).

Exit ticket

Three questions to check you have got it.

What number makes $x^2 + 10x + \;?$ a perfect square?
What number makes $x^2 + 18x + \;?$ a perfect square?
What number makes $x^2 + 4x + \;?$ a perfect square?

Summary card

Completing the square

Add $(b/2)^2$ to both sides; then square roots.

Watch out: halve, then square; and do it on both sides.

End of the lesson

Next lesson: The quadratic formula and the discriminant Back to the chapter

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