Solving Quadratic Equations by Completing the Square 

Some quadratic equations cannot be readily factored and aren't given in a format that allows us to use the square root property immediately.  However, we can use a technique called "completing the square" to rewrite the quadratic expression as a perfect square trinomial. We can then factor the trinomial and solve the equation using the square root property.

Steps to Solving Equations by Completing the Square

1.  Rewrite the equation in the form x² + bx = c.
2.  Add to both sides the term needed to complete the square.
3.  Factor the perfect square trinomial.
4.  Solve the resulting equation by using the square root property.

Finding the Term Needed to Complete the Square

If x² + bx is a binomial, then adding  will result in a perfect square trinomial.   is the square of half the coefficient of the linear x.  

A perfect square trinomial can be factored, so the equation can then be solved by taking the square root of both sides.


Example

Solve the equation  x² + 8x + 5 = 0  by completing the square.

Solution

First, rewrite the equation in the form x² + bx = c.

x² + 8x = −5

Add the appropriate constant to complete the square, then simplify.

x² + 8x + 16 = −5 + 16

(x + 4)² = 11

Now solve using the square root method.

Practice

Solve the following equations.

1.  x² − 8x + 13 = 0

2.  x² + 4x − 16 = 0

3.  x² + 16x + 10 = 0

4.  x² + 9x − 5 = 4x − 7

5.  4x² = x + 2

Answers

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