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Completing the Square

General

Completing the Square

A method for solving quadratic equations by rewriting them into a perfect-square-trinomial form, useful both for solving and for deriving the quadratic formula itself.

Definition

Completing the square is a method for solving a quadratic equation by manipulating it into the form (x + p)² = q, a perfect square trinomial set equal to a constant, which can then be solved by taking a square root. It's one of three standard methods for solving quadratics, alongside factoring and the quadratic formula.

Beyond solving equations, this technique is also how you convert a quadratic from standard form into vertex form, useful for quickly identifying a parabola's maximum or minimum point without graphing it. It's also the method used to derive the quadratic formula itself.

Formula

For ax² + bx + c = 0 (assuming a = 1 for simplicity):

  1. Move c to the other side: x² + bx = −c
  2. Add (b/2)² to both sides: x² + bx + (b/2)² = −c + (b/2)²
  3. Factor the left side as a perfect square: (x + b/2)² = −c + (b/2)²
  4. Take the square root and solve for x

Worked Example

Solve x² + 6x − 7 = 0 by completing the square:

  1. Move constant: x² + 6x = 7
  2. Add (6/2)² = 9 to both sides: x² + 6x + 9 = 16
  3. Factor: (x + 3)² = 16
  4. Take square root: x + 3 = ±4
  5. Solve: x = 1 or x = −7

Checking against the original equation confirms both values satisfy x² + 6x − 7 = 0.

Key Things to Know

  • Divide by the leading coefficient first if it isn't 1. Skipping this step breaks the perfect-square pattern and leads to an incorrect result.
  • The value added to both sides is always (b/2)². This is the key step that turns the left side into a factorable perfect square trinomial.
  • Useful beyond just solving, for finding a parabola's vertex. Converting to (x + p)² = q form directly reveals the vertex coordinates without needing calculus or graphing.
  • Mathematically identical to using the quadratic formula. Both methods always agree, completing the square is simply the derivation path that produces the quadratic formula in general form.
  • Works even when the equation has no real solutions. You'll encounter a negative number under a square root partway through, revealing complex solutions rather than a dead end.

Related Calculators

Quadratic Formula Calculator →Solve any quadratic equation instantly. Enter coefficients a, b, and c to find real and complex roots with step-by-step working shown for free.

Related Terms

Quadratic EquationQuadratic EquationDiscriminantDiscriminant (b² − 4ac)

Frequently Asked Questions

Why learn completing the square if the quadratic formula already solves any equation?
Completing the square builds the intuition behind where the quadratic formula actually comes from, and it's also directly useful outside solving equations, for converting a parabola's equation into vertex form to quickly identify its highest or lowest point.
What happens if the leading coefficient isn't 1?
You need to divide the entire equation by that coefficient first, before completing the square, otherwise the perfect-square-trinomial pattern won't work out correctly.
Is completing the square faster than factoring?
Not usually for equations that factor cleanly, factoring is quicker there. Completing the square shines when factoring isn't obvious or the equation doesn't factor into nice integers at all.
Does completing the square always give the same answer as the quadratic formula?
Yes, they're mathematically equivalent methods, completing the square is actually how the quadratic formula itself is derived, so both approaches always produce identical solutions.
Can completing the square handle equations with no real solutions?
Yes, it'll lead you to taking the square root of a negative number partway through, revealing that the solutions are complex, the same information the discriminant would tell you upfront.