Solve a quadratic by completing the square
Completing the square rewrites a quadratic so one side becomes a perfect square trinomial. This makes it possible to isolate the variable by taking a square root.
Method
- Put the equation in standard form if needed: move the constant term to the other side so the quadratic and linear terms stay together.
- Make the leading coefficient 1 on the squared term. If it is not 1, divide every term by that coefficient first.
- Move the constant term to the opposite side.
- Complete the square: take half of the coefficient of the linear term, square it, and add that number to both sides.
- Factor the perfect square trinomial into a squared binomial.
- Take the square root of both sides. Remember to include both the positive and negative square roots.
- Solve for the variable and simplify the answers.
Check your work
Substitute each solution back into the original equation. A correct answer will make the left side equal the right side. If you get a square root of a negative number while working over the real numbers, there is no real solution.
Tip
Keep the same operation on both sides at every step. Completing the square is most reliable when you work carefully with fractions and signs.