Chapter 12.1, Problem 38WE

To determine

To solve: The equation ${\left(y-\frac{3}{7}\right)}^2=-\frac{8}{9}$ and express irrational solution in simplest radical form.

Answer to Problem 38WE

The solution set is $\left(\frac{9+14\sqrt{2}}{21},\frac{9+14\sqrt{2}}{21}\right)$ .

Explanation of Solution

Given information:

The given equation ${\left(y-\frac{3}{7}\right)}^2=-\frac{8}{9}$ .

Calculation:

  $\begin{array}{l}\Rightarrow {\left(y-\frac{3}{7}\right)}^2=-\frac{8}{9}\\ \Rightarrow y-\frac{3}{7}=\pm \sqrt{-\frac{8}{9}}\\ \Rightarrow y-\frac{3}{7}=\pm \left(-\frac{2\sqrt{2}}{3}\right)\\ \Rightarrow y-\frac{3}{7}+\frac{3}{7}=\pm \left(-\frac{2\sqrt{2}}{3}\right)+\frac{3}{7}\end{array}$

(Adding $\frac{3}{7}$ on both sides)

  $\begin{array}{l}\Rightarrow y=\pm \left(-\frac{2\sqrt{2}}{3}\right)+\frac{3}{7}\\ \Rightarrow y=\pm \left(\frac{9-14\sqrt{2}}{21}\right)\\ \Rightarrow y=\frac{9+14\sqrt{2}}{21};y=-\frac{9+14\sqrt{2}}{21}\end{array}$

Therefore, the solution set is $\left(\frac{9+14\sqrt{2}}{21},\frac{9+14\sqrt{2}}{21}\right)$ .