Completing the Square

≈ 35 min

Completing the square rewrites $x^2 + bx$ as a perfect bracketed square (plus a correction), so that the square-roots method applies to ANY quadratic.

The recipe: take half of $b$ , square it, add and subtract it.

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

Worked example. Solve $x^2 + 6x + 5 = 0$ .

Half of 6 is 3; $3^2 = 9$ :

x^2 + 6x + 9 - 9 + 5 = 0 \;\Rightarrow\; (x + 3)^2 - 4 = 0

(x + 3)^2 = 4 \;\Rightarrow\; x + 3 = \pm 2

x = -1 \quad \text{or} \quad x = -5

Check: $(-1)^2 + 6(-1) + 5 = 0$ ✓ and $(-5)^2 + 6(-5) + 5 = 0$ ✓

What number must be added to $x^2 + 6x$ to complete the square?

Solve by completing the square: $x^2 + 6x + 5 = 0$ . Answer in the form x=… or x=…

To complete the square on $x^2 - 10x$ , you add…

$100$ , because $10^2 = 100$

$25$ , because $(-10 \div 2)^2 = 25$

$5$ , because half of 10 is 5

$-25$ , because $b$ is negative

Write $x^2 + 4x + 1$ in completed-square form. Answer like (x+2)^2-3.