Learning Objectives

In this section you will:

5.5.1 – Completing the Square

Not all quadratic equations can be factored or can be solved in their original form using the square root property. In these cases, we may use a method for solving a quadratic equation known as completing the square. Using this method, we add or subtract terms to both sides of the equation until we have a perfect square trinomial on one side of the equal sign. We then apply the square root property. To complete the square, the leading coefficient, a, must equal 1. If it does not, then divide the entire equation by a. Then, we can use the following procedures to solve a quadratic equation by completing the square.

We will use the example $\,{x}^{2}+4x+1=0\,$ to illustrate each step.

1. Given a quadratic equation that cannot be factored, and with $\,a=1,$ first add or subtract the constant term to the right sign of the equal sign.

${x}^{2}+4x=-1$
2. Multiply the b term by $\,\frac{1}{2}\,$ and square it.

$\begin{array}{ccc}\hfill \frac{1}{2}\left(4\right)& =& 2\hfill \\ \hfill {2}^{2}& =& 4\hfill \end{array}$
3. Add $\,{\left(\frac{1}{2}b\right)}^{2}\,$ to both sides of the equal sign and simplify the right side. We have

$\begin{array}{ccc}\hfill {x}^{2}+4x+4& =& -1+4\hfill \\ \hfill {x}^{2}+4x+4& =& 3\hfill \end{array}$
4. The left side of the equation can now be factored as a perfect square.

$\begin{array}{ccc}\hfill {x}^{2}+4x+4& =& 3\hfill \\ \hfill {\left(x+2\right)}^{2}& =& 3\hfill \end{array}$
5. Use the square root property and solve.

$\begin{array}{ccc}\hfill \sqrt{{\left(x+2\right)}^{2}}& =& ±\sqrt{3}\hfill \\ \hfill x+2& =& ±\sqrt{3}\hfill \\ \hfill x& =& -2±\sqrt{3}\hfill \end{array}$
6. The solutions are $\,-2+\sqrt{3},$ $\text{and}-2-\sqrt{3}.$

Example 1 – Solving a Quadratic by Completing the Square

Solve the quadratic equation by completing the square: $\,{x}^{2}-3x-5=0.$

First, move the constant term to the right side of the equal sign.

${x}^{2}-3x=5$

Then, take $\,\frac{1}{2}\,$ of the b term and square it.

$\begin{array}{ccc}\hfill \frac{1}{2}\left(-3\right)& =& -\frac{3}{2}\hfill \\ \hfill {\left(-\frac{3}{2}\right)}^{2}& =& \frac{9}{4}\hfill \end{array}$

Add the result to both sides of the equal sign.

$\begin{array}{ccc}\hfill {x}^{2}-3x+{\left(-\frac{3}{2}\right)}^{2}& =& 5+{\left(-\frac{3}{2}\right)}^{2}\hfill \\ \hfill {x}^{2}-3x+\frac{9}{4}& =& 5+\frac{9}{4}\hfill \end{array}$

Factor the left side as a perfect square and simplify the right side.

${\left(x-\frac{3}{2}\right)}^{2}=\frac{29}{4}$

Use the square root property and solve.

$\begin{array}{ccc}\hfill \sqrt{{\left(x-\frac{3}{2}\right)}^{2}}& =& \pm\sqrt{\frac{29}{4}}\hfill \\ \hfill \left(x-\frac{3}{2}\right)& =& \pm\frac{\sqrt{29}}{2}\hfill \\ \hfill x& =& \frac{3}{2}\pm\frac{\sqrt{29}}{2}\hfill \end{array}$

The solutions are $\,\frac{3}{2}+\frac{\sqrt{29}}{2},$ $\text{and} \frac{3}{2}-\frac{\sqrt{29}}{2}.$

Try It

Solve by completing the square: $\,{x}^{2}-6x=13.$

$x=3\pm\sqrt{22}$

Try It

Solve by completing the square: $\,{x}^{2}-8x=-6.$

$x=4\pm\sqrt{10}$

Try It

Solve by completing the square: $\,{x}^{2}+4x-10=0.$

$x=-2\pm\sqrt{14}$

Try It

Solve by completing the square: $\,2{x}^{2}-12x+9=0.$

$x=3\pm\frac{3}{\sqrt{2}}$

Access these online resources for additional instruction and practice with quadratic equations.

Key Equations

 quadratic formula $x=\frac{-b±\sqrt{{b}^{2}-4ac}}{2a}$

Key Concepts

• Completing the square is a method of solving quadratic equations when the equation cannot be factored. See (Example 1)

Glossary

completing the square
a process for solving quadratic equations in which terms are added to or subtracted from both sides of the equation in order to make one side a perfect square
discriminant
the expression under the radical in the quadratic formula that indicates the nature of the solutions, real or complex, rational or irrational, single or double roots.
Pythagorean Theorem
a theorem that states the relationship among the lengths of the sides of a right triangle, used to solve right triangle problems
one of the methods used to solve a quadratic equation, in which the $\,{x}^{2}\,$ term is isolated so that the square root of both sides of the equation can be taken to solve for x