Quadratic equations can be solved using a variety of strategies. One strategy is to solve by a combination of factoring and square roots. Look at the equation below.

x2 + 12x + 36 = 49

The left side of the equation is a perfect square trinomial and can be factored as shown.

(x + 6)2 = 49

To solve for x, take the square root of each side of the equation.

√(x + 6)2 = √49


x + 6 = ±7


x + 6 = 7
x = 1

or

x + 6 = -7
x = -13

If one side of the equation is not a perfect square trinomial, you can rewrite it by using a strategy called 'completing the square.' Look at the example below.

x2 – 8x – 9 = 0
x2 – 8x = 9

The left side of the equation is not a perfect square trinomial because it cannot be factored as a square of a sum or difference of two terms.

Move the constant to the right-hand side of the equation.

Find a constant for the left-hand side of the equation that would create a perfect square trinomial.

(numerator: -8, demoniator: 2) squared = (-4)2 = 16

x2 –8x + 16 = 9 + 16

(x – 4)2 = 25


√(x − 4)2 = √25

x – 4 = ±5

First, divide b (b = -8) by 2, and then square the result.

Add this to both sides of the equation to keep the equation balanced.

Factor the left side of the equation and simplify the right side of the equation.


Take the square root of each side of the equation.

Solve for x.

x – 4 = 5

x = 9

or

x – 4 = -5

x = -1

The solutions to the equation are 9 and -1.

This method can also be used when the value of a > 1.

Simply divide the entire equation by the value of a before beginning the process of completing the square.

2x2 + 24x – 28 = 0

2x2 + 24x – 28 2 = 0 2

x2 + 12x – 14 = 0

x2 + 12x = 14

x2 + 12x + 36 = 14 + 36

(x + 6)2 = 50

√(x + 6)2 = √50

x + 6 ≈ ±7.07

x + 6 ≈ 7.07

x ≈ 1.07


or

x + 6 ≈ -7.07

x ≈ -13.07

Round the answer to nearest hundredths place.

If you would like to look at another example and/or work some problems on your own, click here.