9.5 Completing the Square

Notes

Completing the square turns $x^2 + bx$ into a perfect square by adding $\displaystyle \left(\frac{b}{2}\right)^2$. 1. Get $x^2 + bx$ alone on one side (divide by $a$ first if needed). 2. Add $\displaystyle \left(\frac{b}{2}\right)^2$ to both sides. 3. Write the left side as $\displaystyle \left(x + \frac{b}{2}\right)^2$ and solve with square roots. Example: $x^2 + 6x = 7 \Rightarrow x^2 + 6x + 9 = 16 \Rightarrow (x + 3)^2 = 16 \Rightarrow x = 1$ or $x = -7$. It also converts standard form to vertex form.