91Ó°ÊÓ

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

$$\begin{gathered} n^2−7n=5 \\ {n}^2−7n=5\text{ â¶Ä‰â¶Ä‰â¶Ä‰}[\text{ â¶Ä‰S¾±²Ô³¦±ð â¶Ä‰}{\left(\frac{−7}{2}\right)}^2=\frac{49}{4}\text{ â¶Ä‰},\text{ add â¶Ä‰}\frac{49}{4}\text{ â¶Ä‰to â¶Ä‰each â¶Ä‰side â¶Ä‰}] \\ {n}^2−7n+\frac{49}{4}=5+\frac{49}{4} \end{gathered}$$

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

role="math" localid="1647849647247" $$\begin{gathered} {\left(n−\frac{7}{2}\right)}^2=\frac{69}{4} \\ \sqrt{{(n−\frac{7}{2})}^2}=Â\pm \sqrt{\frac{69}{4}}\text{ â¶Ä‰â¶Ä‰â¶Ä‰â€‰â¶Ä‰â¶Ä‰â¶Ä‰â€‰}\left[\text{ â¶Ä‰Take â¶Ä‰the â¶Ä‰square â¶Ä‰root â¶Ä‰of â¶Ä‰each â¶Ä‰side â¶Ä‰â€‰}\right] \\ n−\frac{7}{2}=Â\pm \frac{\sqrt{69}}{2} \end{gathered}$$

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

$$\begin{gathered} n−\frac{7}{2}+\frac{7}{2}=\frac{7}{2}Â\pm \frac{\sqrt{69}}{2} \\ n=\frac{7}{2}−\frac{\sqrt{69}}{2}\text{ â¶Ä‰â¶Ä‰â¶Ä‰â€‰or â¶Ä‰â€‰}n\text{ â¶Ä‰}=\text{ }\frac{7}{2}\text{ â¶Ä‰}+\text{ â¶Ä‰â¶Ä‰â¶Ä‰}\frac{\sqrt{69}}{2} \end{gathered}$$

$$\begin{gathered} n=0.653\text{ â¶Ä‰â¶Ä‰â¶Ä‰or â¶Ä‰}n\text{ â¶Ä‰}=\text{ â¶Ä‰}7.653\text{ } \\ n≈0.7\text{ â¶Ä‰â¶Ä‰â¶Ä‰or â¶Ä‰â¶Ä‰â¶Ä‰}n≈7.7 \end{gathered}$$

Therefore, the solutions are 0.7 and 7.7.