Question

Solve each equation by completing the square. Round to the nearest tenth if necessary

$n^2-7n=5$

Short answer

Thesolutions of the quadratic equations are 0.7 and 7.7.

Step-by-step solution

Step 1. Complete the square.

Complete the square to solve the quadratic equations by isolating$x^2$ andbx-terms.

$$\begin{gathered} n^2-7n=5 \\ {n}^2-7n=5[\text{\hspace{0.17em}\hspace{0.17em}Since\hspace{0.17em}\hspace{0.17em}}{\left(\frac{-7}{2}\right)}^2=\frac{49}{4},\text{\hspace{0.17em}add\hspace{0.17em}\hspace{0.17em}}\frac{49}{4}\text{\hspace{0.17em}\hspace{0.17em}to\hspace{0.17em}\hspace{0.17em}each\hspace{0.17em}\hspace{0.17em}side\hspace{0.17em}\hspace{0.17em}}] \\ {n}^2-7n+\frac{49}{4}=5+\frac{49}{4} \end{gathered}$$

Step 2. Take the square root of each side. 

The first factor $n^2-7n+\frac{49}{4}$, then take the square root of each side.

role="math" localid="1647849647247" $$\begin{gathered} {\left(n-\frac{7}{2}\right)}^2=\frac{69}{4} \\ \sqrt{{(n-\frac{7}{2})}^2}=\pm \sqrt{\frac{69}{4}}\left[\text{\hspace{0.17em}\hspace{0.17em}Take\hspace{0.17em}\hspace{0.17em}the\hspace{0.17em}\hspace{0.17em}square\hspace{0.17em}\hspace{0.17em}root\hspace{0.17em}\hspace{0.17em}of\hspace{0.17em}\hspace{0.17em}each\hspace{0.17em}\hspace{0.17em}side\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\right] \\ n-\frac{7}{2}=\pm \frac{\sqrt{69}}{2} \end{gathered}$$

Step 3. Write the solutions.

Now add $\frac{7}{2}$to both sides and separate the solutions.

$$\begin{gathered} n-\frac{7}{2}+\frac{7}{2}=\frac{7}{2}\pm \frac{\sqrt{69}}{2} \\ n=\frac{7}{2}-\frac{\sqrt{69}}{2}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}or\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}n=\frac{7}{2}+\frac{\sqrt{69}}{2} \end{gathered}$$

$$\begin{gathered} n=0.653\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}or\hspace{0.17em}\hspace{0.17em}}n=7.653 \\ n\approx 0.7\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}or\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}n\approx 7.7 \end{gathered}$$

Therefore, the solutions are 0.7 and 7.7.