Question

Solve $3r^2-2r=21$by completing the square.

Short answer

The values of $r$for the equation $3r^2-2r=21$by completing the squares are$3,-\frac{7}{3}$.

Step-by-step solution

Step 1. Given Information.

The equation is,$3r^2-2r=21$.

Step 2. Using completing the square method.

Given, $3r^2-2r=21$

Dividing both sides by $3$,

$$\begin{gathered} \Rightarrow \frac{3r^2-2r}{3}=\frac{21}{3} \\ \Rightarrow r^2-\frac{2}{3}r=7 \end{gathered}$$

Taking the square of the half of $-\frac{2}{3}$and adding it on both sides,

$$\begin{gathered} \Rightarrow r^2+{\left(\frac{1}{2}\left(-\frac{2}{3}\right)\right)}^2-\frac{2}{3}r=7+{\left(\frac{1}{2}\left(-\frac{2}{3}\right)\right)}^2 \\ \Rightarrow r^2+\frac{1}{9}-\frac{2}{3}r=7+\frac{1}{9} \\ \Rightarrow r^2-\frac{2}{3}r+\frac{1}{9}=\frac{64}{9} \end{gathered}$$

Factoring the right side as a binomial square,

$\Rightarrow {\left(r-\frac{1}{3}\right)}^2=\frac{64}{9}$

Taking the square root on both sides,

$$\begin{gathered} \Rightarrow r-\frac{1}{3}=\sqrt{\frac{64}{9}} \\ \Rightarrow r-\frac{1}{3}=\pm \frac{8}{3} \end{gathered}$$

Solving for $r$,

$$\begin{gathered} \Rightarrow r=\frac{1}{3}\pm \frac{8}{3} \\ \Rightarrow r=\frac{1}{3}+\frac{8}{3} \\ or,r=\frac{1}{3}-\frac{8}{3} \\ \Rightarrow r=3,-\frac{7}{3} \end{gathered}$$