Completing the Square: Step-by-Step Methodology in 2026
Completing the square is an algebraic technique that rewrites any quadratic equation $ax^2 + bx + c = 0$ into a perfect square binomial form $a(x - h)^2 + k = 0$. It is the mathematical derivation behind the Quadratic Formula.
Step-by-Step Computational Workflow
- 1Divide by $a$: Standardize equation to monic form: $x^2 + \frac{b}{a}x = -\frac{c}{a}$.
- 2Find the Magic Term: Halve the linear coefficient and square it: $\left(\frac{b}{2a}\right)^2$.
- 3Add Term to Both Sides: $x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \left(\frac{b}{2a}\right)^2$.
- 4Factor the Left Side: $\left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2}$.
- 5Solve via Square Root: $x = -\frac{b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a}$.