# Complete the Square to Solve Quadratic Equations

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Examples, videos and solutions to help GCSE Maths students learn how to complete the square in order to solve quadratic equations.

How to complete the square when solving quadratic equations:

Completing the square means re-arranging the quadratic equation into “perfect square” form (x + p)2 = q, or (x - p)2 = q. Then, we can solve by square-rooting both sides of the equation.

1. x2 (for ex: x2 + 6x - 7 = 0), move constant to right, add (6/2)2 to both sides, write left side as perfect square, and square root both sides to solve. Remember the plus and minus solutions when you square root the constant side.

2. 2x2 or 3x2 or 4x2 etc. (for ex: 2x2 - 10x - 3 = 0), move constant to right side, divide each term by leading coefficient 2 on left and right side, add (5/2)2 to both sides, write left side as perfect square and square root both sides to solve.

3. -x2 or any negative coefficient (for ex: -x2 - 6x + 7 = 0), move constant to right side, divide by -1 on left and right side. (NOTE: if you had -2x2 in your equation, you would divide every term by -2 on left and right side). Then add (6/2)2 to both sides, write left side as perfect square and square root both sides to solve.

If you don’t have an X term in equation, (for ex: 3x2 - 121 = 0), then you cannot complete the square. Just move the constant to the right, divide both sides by 3, and square root both sides to solve. Remember plus and minus solutions on the right.

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations. 