Making a perfect square
For \(x^2+bx\), halve the coefficient of \(x\), then square it. Adding and subtracting the same number preserves the expression.
\(\begin{aligned} x^2+bx+c&=x^2+bx+\left(\dfrac b2\right)^2+c-\left(\dfrac b2\right)^2 \\ &=\left(x+\dfrac b2\right)^2+c-\dfrac{b^2}{4} \end{aligned}\)
For example, \(x^2-2x-3=(x-1)^2-4\). The graph has its minimum at \((1,-4)\). Its roots are the \(x\)-coordinates where it meets the \(x\)-axis.
Solving the completed square
| Completed equation | Real solutions |
|---|---|
| \((x-h)^2=k\), where \(k>0\) | \(x=h+\sqrt{k}\) or \(x=h-\sqrt{k}\) |
| \((x-h)^2=0\) | \(x=h\) |
| \((x-h)^2=k\), where \(k<0\) | No real solution: the square of a real number cannot be negative. |
Keep both square-root branches. Use an exact value until the final rounding step.
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