How to Solve by Completing the Square

Completing the square turns x² + bx into a perfect square like (x − 3)². Then you can solve by taking square roots. It also shows the vertex of a parabola.

Try it with your own problem ↓

The steps

Make sure the x² coefficient is 1 (divide everything by it if not).
Move the constant to the right side.
Take half of the x coefficient, square it, and add it to both sides.
Factor the left side as a perfect square: (x + half)².
Take the square root of both sides (± on the right), then solve for x.

A worked example

Solve x² − 6x + 2 = 0.

Move the constant: x² − 6x = −2.
Half of −6 is −3, and (−3)² = 9. Add 9 to both sides: x² − 6x + 9 = −2 + 9 = 7.
The left side factors: (x − 3)² = 7.
Take the square root of both sides: x − 3 = ±√7.
Add 3: x = 3 ± √7.

x = 3 ± √7, which is about 5.646 and 0.354.

Try it with your own problem

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Work it out: solve x^2-6x+2=0 by completing the squareOpens The Area Box with this example

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Common mistakes

Adding the square to only one side. Add it to both sides to keep the equation balanced.
Using the full coefficient instead of half. Half of −6 is −3, not −6.
Forgetting the ±. A square root gives two answers.
Sign slip in the factor. The sign inside matches the sign of the x term: x² − 6x + 9 = (x − 3)².

Questions people ask

Why does it work?

(x − 3)² expands to x² − 6x + 9. Adding 9 gives the missing piece that makes a perfect square.

What if the x² coefficient isn't 1?

Divide the whole equation by it first. For 2x² − 8x + 3, factor 2 out of the x terms.

How does it help find a vertex?

y = a(x − h)² + k has its vertex at (h, k). For y = 2x² − 8x + 3 this gives 2(x − 2)² − 5, so the vertex is (2, −5).

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