Chapter 12.2, Problem 17WE

a.

To determine

To Solve: The equation $2y^2=9y-9$ by completing the square.

Answer to Problem 17WE

The required solution set is $\left\{3,\frac{3}{2}\right\}$ .

Explanation of Solution

Given information:

  $2y^2=9y-9$

Calculation:

  $\begin{array}{l}2y^2=9y-9\\ 2y^2-9y=-9\\ y^2-\frac{9}{2}y=\frac{-9}{2}\end{array}$

(Divide both sides by 2 so that the coefficient of $y^2$ will be 1)

  $y^2-\frac{9}{2}y+{\left(-\frac{9}{4}\right)}^2=\left(\frac{-9}{2}\right)+{\left(-\frac{9}{4}\right)}^2$

(Add square of half the coefficient of y to both sides)

  $\begin{array}{l}{y}^2-\frac{9}{2}y+\left(\frac{81}{16}\right)=\left(\frac{-9}{2}\right)+\left(\frac{81}{16}\right)\\ {\left(y-\frac{9}{4}\right)}^2=\frac{9}{16}\end{array}$

(Simplify)

  $\begin{array}{l}y-\frac{9}{4}=\sqrt{\frac{9}{16}}\\ y-\frac{9}{4}=\pm \frac{3}{4}\end{array}$

(Perform square root on both sides)

  $y=\frac{9}{4}\pm \frac{3}{4}$

Therefore,

  $\begin{array}{l}y=\frac{9}{4}+\frac{3}{4}\\ y=\frac{12}{4}\\ y=3\end{array}$

Or

  $\begin{array}{l}y=\frac{9}{4}-\frac{3}{4}\\ y=\frac{6}{4}\\ y=\frac{3}{2}\end{array}$

Therefore, the solution set is $\left\{3,\frac{3}{2}\right\}$ .

b.

To determine

To Solve: The equation $2y^2=9y-9$ by factoring.

Answer to Problem 17WE

The required solution set is $\left\{3,\frac{3}{2}\right\}$

Explanation of Solution

Given information:

  $2y^2=9y-9$

Calculation:

  $\begin{array}{l}2y^2=9y-9\\ 2y^2-9y+9=0\\ 2y^2-6y-3y+9=0\\ 2y\left(y-3\right)-3\left(y-3\right)=0\\ \left(y-3\right)\left(2y-3\right)\end{array}$

(Splitting the middle term and factoring)

So,

  $\begin{array}{l}y-3=0\\ y=3\end{array}$

Or

  $\begin{array}{l}2y-3=0\\ y=\frac{3}{2}\end{array}$

Therefore, the solution set is $\left\{3,\frac{3}{2}\right\}$ .