Chapter 12.2, Problem 16WE

a.

To determine

To Solve: The equation $3c^2-7c=6$ by completing the square.

Answer to Problem 16WE

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

Explanation of Solution

Given information:

  $3c^2-7c=6$

Calculation:

  $\begin{array}{l}3c^2-7c=6\\ c^2-\frac{7c}{3}=2\end{array}$

(Divide each side by 3 so that the coefficient of $c^2$ will be 1)

  $c^2-\frac{7c}{3}+{\left(-\frac{7}{6}\right)}^2=2+{\left(-\frac{7}{6}\right)}^2$

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

  $\begin{array}{l}{c}^2-\frac{7c}{3}+\left(\frac{49}{36}\right)=2+\left(\frac{49}{36}\right)\\ {\left(c-\frac{7}{6}\right)}^2=\frac{121}{36}\end{array}$

(Simplify)

  $c-\frac{7}{6}=\pm \frac{11}{6}$

(Perform square root on both sides)

  $c=\frac{7}{6}\pm \frac{11}{6}$

Therefore,

  $\begin{array}{l}c=\frac{7}{6}+\frac{11}{6}\\ c=\frac{7+11}{6}\\ c=3\end{array}$

Or

  $\begin{array}{l}c=\frac{7}{6}-\frac{11}{6}\\ c=\frac{7-11}{6}\\ c=\frac{-2}{3}\end{array}$

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

b.

To determine

To Solve: The equation $3c^2-7c=6$ by factoring.

Answer to Problem 16WE

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

Explanation of Solution

Given information:

  $3c^2-7c=6$

Calculation:

  $\begin{array}{l}3c^2-7c=6\\ 3c^2-7c-6=0\\ 3c^2-9c+2c-6=0\\ 3c\left(c-3\right)+2\left(c-3\right)=0\\ \left(c-3\right)\left(3c+2\right)=0\end{array}$

(Splitting the middle term and factoring)

Therefore,

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

Or

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

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