Lesson 13

Completing the Square (Part 2)

• Let’s solve some harder quadratic equations.

13.1: Math Talk: Equations with Fractions

Solve each equation mentally.

$$x+x=\frac14$$

$$(\frac32) ^2 = x$$

$$\frac35 + x = \frac95$$

$$\frac{1}{12}+x=\frac14$$

13.2: Solving Some Harder Equations

Solve these equations by completing the square.

1. $$(x-3)(x+1)=5$$
2. $$x^2 + \frac12 x = \frac{3}{16}$$
3. $$x^2+3x+\frac84=0$$
4. $$(7-x)(3-x)+3=0$$
5. $$x^2+1.6x+0.63=0$$

1. Show that the equation $$x^2+10x+9=0$$ is equivalent to $$(x+3)^2+4x=0$$.
2. Write an equation that is equivalent to $$x^2+9x+16=0$$ and that includes $$(x+4)^2$$.

13.3: Spot Those Errors!

Here are four equations, followed by worked solutions of the equations. Each solution has at least one error.

• Solve one or more of these equations by completing the square.
• Then, look at the worked solution of the same equation as the one you solved. Find and describe the error or errors in the worked solution.
1. $$x^2 + 14x= \text-24$$
2. $$x^2 - 10x + 16= 0$$
3. $$x^2 + 2.4x = \text-0.8$$
4. $$x^2 - \frac65 x + \frac15 = 0$$

Worked solutions (with errors):

1.

\displaystyle \begin {align} x^2 + 14x &= \text-24\\ x^2 + 14x + 28 &= 4\\ (x+7)^2 &= 4\\ \\x+7 = 2 \quad &\text {or} \quad x+7 = \text-2\\ x = \text-5 \quad &\text {or} \quad x = \text-9 \end{align}

2.

\displaystyle \begin {align} x^2 - 10x + 16 &= 0\\x^2 - 10x + 25 &= 9\\(x - 5)^2 &= 9\\ \\x-5=9 \quad &\text {or} \quad x-5 = \text-9\\ x=14 \quad &\text {or} \quad x=\text-4 \end{align}

3.

\displaystyle \begin {align}x^2 + 2.4x &= \text-0.8\\x^2 + 2.4x + 1.44 &= 0.64\\(x + 1.2)^2&=0.64\\x+1.2 &= 0.8\\ x &=\text -0.4 \end{align}

4.

\displaystyle \begin {align} x^2 - \frac65 x + \frac15 &= 0\\x^2 - \frac65 x + \frac{9}{25} &= \frac{9}{25}\\ \left(x-\frac35\right)^2 &= \frac{9}{25}\\ \\x-\frac35= \frac35 \quad &\text {or} \quad x-\frac35=\text- \frac35\\ x=\frac65 \quad &\text {or} \quad x=0 \end{align}

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:

$$\displaystyle x^2 + 5x - \frac{75}{4}=0$$

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

\displaystyle \begin {align} x^2 + 5x - \frac{75}{4}+ \frac{75}{4} &= 0+\frac{75}{4}\\ x^2 + 5x &= \frac{75}{4} \end {align}

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

\displaystyle \begin {align}x^2 + 5x + \frac{25}{4} &= \frac{75}{4} + \frac{25}{4}\\x^2 + 5x + \frac{25}{4} &= \frac{100}{4} \end{align}

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

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

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

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

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

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

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

\displaystyle \begin {align} 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{align}

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

Glossary Entries

• completing the square

Completing the square in a quadratic expression means transforming it into the form $$a(x+p)^2-q$$, where $$a$$, $$p$$, and $$q$$ are constants.

Completing the square in a quadratic equation means transforming into the form $$a(x+p)^2=q$$.

• perfect square

A perfect square is an expression that is something times itself. Usually we are interested in situations where the something is a rational number or an expression with rational coefficients.

• rational number

A rational number is a fraction or the opposite of a fraction. Remember that a fraction is a point on the number line that you get by dividing the unit interval into $$b$$ equal parts and finding the point that is $$a$$ of them from 0. We can always write a fraction in the form $$\frac{a}{b}$$ where $$a$$ and $$b$$ are whole numbers, with $$b$$ not equal to 0, but there are other ways to write them. For example, 0.7 is a fraction because it is the point on the number line you get by dividing the unit interval into 10 equal parts and finding the point that is 7 of those parts away from 0. We can also write this number as $$\frac{7}{10}$$.

The numbers $$3$$, $$\text-\frac34$$, and $$6.7$$ are all rational numbers. The numbers $$\pi$$ and $$\text-\sqrt{2}$$ are not rational numbers, because they cannot be written as fractions or their opposites.