What Is Completing the Square?
Completing the square is an algebraic technique used to solve quadratic equations, rewrite quadratic expressions in vertex form, and derive the quadratic formula. The method transforms ax² + bx + c = 0 into a perfect square trinomial on one side of the equation.
The Method Step-by-Step
Step 1: Divide by a
If a is not 1, divide every term by a to make the leading coefficient 1.
x² + (b/a)x + (c/a) = 0
Step 2: Move Constant
Move the constant term to the right side.
x² + (b/a)x = -(c/a)
Step 3: Complete the Square
Add (b/2a)² to both sides.
(x + b/2a)² = (b/2a)² - c/a
Step 4: Solve
Take the square root of both sides and solve for x.
x = -b/2a +/- sqrt((b/2a)² - c/a)
When to Use This Method
- Converting a quadratic to vertex form for graphing.
- Solving quadratics when factoring is not straightforward.
- Deriving the quadratic formula itself.
- Simplifying integrals involving quadratic expressions in calculus.
Relationship to the Quadratic Formula
The quadratic formula x = (-b +/- sqrt(b² - 4ac)) / 2a is actually derived by completing the square on the general form ax² + bx + c = 0. Understanding completing the square gives deeper insight into why the quadratic formula works.