Complete the Square to Solve Quadratic Equations

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.

If your quadratic equation starts with:

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.

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