Algebra Bridge
Completing the Square
Completing the square rewrites a quadratic expression as a perfect square plus or minus a constant. It is the bridge from quadratic formulas to vertex form, circles, and conic sections.
Fact Table
| Goal | Move | Result |
|---|---|---|
| Turn $x^2+bx$ into a square. | Add $\left(\frac b2\right)^2$. | $x^2+bx+\left(\frac b2\right)^2=\left(x+\frac b2\right)^2$ |
| Leading coefficient is not 1. | Factor $a$ from the quadratic and linear terms first. | Complete the square inside parentheses. |
| Equation must stay balanced. | Add the same value to both sides. | No solution set changes. |
| Expression is being rewritten. | Add and subtract the same value. | The expression stays equivalent. |
Content Formulas
Square Add-On
$$\left(\frac b2\right)^2$$
Perfect Square
$$x^2+bx+\left(\frac b2\right)^2=\left(x+\frac b2\right)^2$$
For conics, group x-terms together and y-terms together before completing each square.
Classic Examples
Rewrite in Vertex Form
Write $y=x^2+6x+2$ in vertex form.
Solution StrategyAdd and subtract the square of half the linear coefficient.
$$y=x^2+6x+2$$
$$y=x^2+6x+9-9+2$$
$$y=(x+3)^2-7$$
$$\text{vertex: }(-3,-7)$$
Leading Coefficient
Write $y=2x^2-12x+5$ in vertex form.
Solution StrategyFactor $2$ from the x-terms first, then complete the square inside.
$$y=2(x^2-6x)+5$$
$$y=2(x^2-6x+9-9)+5$$
$$y=2(x-3)^2-18+5$$
$$y=2(x-3)^2-13$$