4.3 Completing the Square

Notes

Completing the square turns $x^2 + bx$ into a perfect square by adding $\displaystyle \left(\frac{b}{2}\right)^2$.

• To solve $x^2 + bx + c = 0$: move $c$, add $\displaystyle \left(\frac{b}{2}\right)^2$ to both sides, factor, then use square roots.
• Example: $x^2 + 6x - 7 = 0 \Rightarrow x^2 + 6x = 7 \Rightarrow x^2 + 6x + 9 = 16 \Rightarrow (x + 3)^2 = 16 \Rightarrow x = 1$ or $x = -7$.
• If $a \neq 1$, divide every term by $a$ first.