Question

Solve by completing the square:

$3q^2-5q=9$.

Short answer

The values of $q$for the equation $3q^2-5q=9$by completing the square are$\frac{5+\sqrt{133}}{6},\frac{5-\sqrt{133}}{6}$.

Step-by-step solution

Step 1. Given Information.

The equation is,

$3q^2-5q=9$.

Step 2. Completing the square.

Given, $3q^2-5q=9$

Dividing both sides by $3$to remove the coefficient of $q^2$on both sides,

$$\begin{gathered} \Rightarrow \frac{3q^2-5q}{3}=\frac{9}{3} \\ \Rightarrow q^2-\frac{5}{3}q=3 \end{gathered}$$

Adding the square to the half of $(-\frac{5}{3})$on both sides,

$$\begin{gathered} \Rightarrow q^2-\frac{5}{3}q+{\left(\frac{1}{2}\left(-\frac{5}{3}\right)\right)}^2=3+{\left(\frac{1}{2}\left(-\frac{5}{3}\right)\right)}^2 \\ \Rightarrow q^2-\frac{5}{3}q+\frac{25}{36}=3+\frac{25}{36} \\ \Rightarrow q^2-\frac{5}{3}q+\frac{25}{36}=\frac{133}{36} \end{gathered}$$

Taking the left side as the binomial square,

$\Rightarrow {\left(q-\frac{5}{6}\right)}^2=\frac{133}{36}$

Taking the square root on both sides,

$$\begin{gathered} \Rightarrow q-\frac{5}{6}=\sqrt{\frac{133}{36}} \\ \Rightarrow q-\frac{5}{6}=\pm \frac{\sqrt{133}}{6} \end{gathered}$$

Solving for the value of $q$,

$$\begin{gathered} \Rightarrow q=\frac{5}{6}\pm \frac{\sqrt{133}}{6} \\ \Rightarrow q=\frac{5}{6}+\frac{\sqrt{133}}{6} \\ or,q=\frac{5}{6}-\frac{\sqrt{133}}{6} \\ \Rightarrow q=\frac{5+\sqrt{133}}{6},\frac{5-\sqrt{133}}{6} \end{gathered}$$