Completing the Square (Part 2)

Student Summary

Completing the square can be a useful method for solving quadratic equations in cases in which it is not easy to rewrite an expression in factored form. For example, let’s solve this equation:

x2+5x−754=0\displaystyle x^2 + 5x - \frac{75}{4}=0

First, we’ll add 754\frac{75}{4} to each side to make things easier on ourselves.

x2+5x−754+754=0+754x2+5x=754\displaystyle \begin{aligned} x^2 + 5x - \frac{75}{4}+ \frac{75}{4} &= 0+\frac{75}{4}\\ x^2 + 5x &= \frac{75}{4} \end{aligned}

To complete the square, take 12\frac12 of the coefficient of the linear term, 5, which is 52\frac52, and square it, which is 254\frac{25}{4}. Add this to each side:

x2+5x+254=754+254x2+5x+254=1004\displaystyle \begin{aligned}x^2 + 5x + \frac{25}{4} &= \frac{75}{4} + \frac{25}{4}\\x^2 + 5x + \frac{25}{4} &= \frac{100}{4} \end{aligned}

Notice that 1004\frac{100}{4} is equal to 25, and rewrite it:

x2+5x+254=25\displaystyle x^2 + 5x + \frac{25}{4} =25

Since the left side is now a perfect square, let’s rewrite it:

(x+52)2=25\displaystyle \left(x+\frac52 \right)^2 = 25

For this equation to be true, one of these equations must true:

x+52=5orx+52=-5\displaystyle x + \frac52 = 5 \quad \text{or} \quad x + \frac52 = \text-5

To finish up, we can subtract 52\frac52 from each side of the equal sign in each equation.

x=5−52orx=-5−52x=52orx=-152x=212orx=-712\displaystyle \begin{aligned} x = 5 - \frac52 \quad &\text{or} \quad x = \text-5 - \frac52\\x = \frac{5}{2} \quad &\text{or} \quad x = \text-\frac{15}{2}\\x = 2\frac12 \quad &\text{or} \quad x = \text-7\frac12 \end{aligned}

It takes some practice to become proficient at completing the square, but it makes it possible to solve many more equations than we could by methods we learned previously.

Visual / Anchor Chart

Standards

Addressing
A-REI.A

No additional information available.

A-REI.4.b

Solve quadratic equations in one variable.