Completing the square
Completing the square is an algebraic technique that rewrites a quadratic expression x² + bx as a perfect square, (x + b/2)² − (b/2)². It is used to solve quadratic equations, find vertex form, and write equations of circles.
Completing the square is a technique for rewriting a quadratic expression as a perfect square trinomial plus a constant. The key identity: x² + bx becomes (x + b/2)² − (b/2)². You take half the coefficient of x, square it, and add and subtract that value so the expression factors into a squared binomial.
For example, to rewrite x² + 6x + 2, take half of 6 (which is 3), square it (9), and regroup: x² + 6x + 9 − 9 + 2 = (x + 3)² − 7. From this form, solving x² + 6x + 2 = 0 is straightforward: (x + 3)² = 7, so x = −3 ± √7. When the leading coefficient isn't 1, factor it out of the x² and x terms first.
The technique has several important applications. It converts a quadratic from standard form to vertex form, y = a(x − h)² + k, exposing the vertex of the parabola. It turns an expanded circle equation like x² + y² − 4x + 6y = 12 into center-radius form (x − 2)² + (y + 3)² = 25. It is also how the quadratic formula is derived, and in calculus it rewrites integrands so they match inverse trigonometric antiderivative forms.
Completing the square shows up across many exams: AP Calculus AB uses it to prepare integrals for arctangent forms, the FE Mechanical exam applies it to quadratics and conic sections in analytic geometry, and the CLT tests it when converting circle equations to center-radius form.
Key takeaways
- Completing the square rewrites x² + bx as (x + b/2)² − (b/2)² by adding and subtracting the square of half the x-coefficient.
- It solves quadratic equations, converts standard form to vertex form, and rewrites circle equations in center-radius form.
- The quadratic formula is derived by completing the square on ax² + bx + c = 0.
- When the leading coefficient is not 1, factor it out of the x² and x terms before completing the square.
- AP Calculus AB, the FE Mechanical exam, and the CLT all test completing the square in different contexts.