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How to complete the square on a quadratic equation (completing the square).

Updated on July 7, 2010

If you are completing to square all need to do is write the quadratic equation (x²+bx+c) in the form (x+p)² + q. To find p half the coefficient of x and to find q take the square of p and take it off c.

Therefore: p = 1/2b and q = c – p²

So x²+bx+c = (x+b/2)² + c – p²

Example 1 on completing the square.

Write x² +6x + 14 in the form (x+p)² + q.

Note from the quadratic b = 6 and c = 14.

First half the value of b:

½ of 6 = 3

So p = 3

Next square p and take this off c.

q = c – p²

q = 14 - 3²

q = 5

So the completed square form is (x+3)² + 5

Example 2 on completing the square.

Write x² - 8x + 5 in the form (x+p)² + q.

Note from the quadratic b = -8 and c = 5.

First half the value of b:

½ of -8 = -4

So p = -4

Next square p and take this off c.

q = c – p²

q = 5 – (-4)²

q = -11

So the completed square form is (x - 4)² - 11

Example 3 on completing the square.

Write x² + 14x + 29 in the form (x+p)² + q.

Note from the quadratic b = 14 and c = 29.

First half the value of b:

½ of 14 = 7

So p = 7

Next square p and take this off c.

q = c – p²

q = 29 – (7)²

q = -20

So the completed square form is (x + 7)² - 20

For some trickier questions on completing the square then click here.

If you need to see some easier examples on completing the square then click here.

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