Completing the square

$(x + 3)^{2} = 16$ is easy to solve: take the root of both sides. Completing the square rewrites any quadratic so it looks like that.

Why does it work? Picture $x^{2} + 6 x$ as a square plus two strips. Split the $6 x$ into two strips of $3 x$ . Now one small corner is missing, of size $3 \times 3 = 9$ . Add it, and you have a full square.

x^{2} + 6 x + 9 = (x + 3)^{2}

. The 9 completes the square.

Move the constant to the right side.
Take half of $b$ and square it.
Add that number to both sides.
Write the left side as a square.
Take the root of both sides, with $\pm$ .

Work through it

Solve $x^{2} + 6 x - 7 = 0$ .

Move the constant. $x^{2} + 6 x = 7$

Half of 6 is 3, and $3^{2} = 9$ . Add it to both sides. $x^{2} + 6 x + 9 = 16$

Write the square. $(x + 3)^{2} = 16$

Take roots. $x + 3 = \pm 4$

Solve both. $x = 1$ or $x = - 7$

Solve $x^{2} - 4 x = 1$ .

Half of $- 4$ is $- 2$ , and its square is 4. $x^{2} - 4 x + 4 = 5$

Write the square. $(x - 2)^{2} = 5$

Take roots. $x = 2 \pm 5$

Watch out for this

Adding the 9 to the left side only.

It's an equation. What goes on one side goes on the other.

Watch out for this

Forgetting $\pm$ .

$(x + 3)^{2} = 16$ has two roots, because $(- 4)^{2}$ is also 16.

The short version
Take half of

b

, square it, and add that to both sides. The left side becomes a perfect square.

Your turn

Prefer paper? Print the Quadratic graphs and completing the square worksheet, with an answer key.

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Work through it

Solve x2+6x−7=0.

Solve x2−4x=1.

Watch out for this

Watch out for this

Your turn

Solve $x^{2} + 6 x - 7 = 0$ .

Solve $x^{2} - 4 x = 1$ .