Question

Solve by completing the square :

$3y^2-10y=-5$

Short answer

The required value of$y$is$\frac{5+\sqrt{10}}{3},\frac{5-\sqrt{10}}{3}$

Step-by-step solution

Given information

$3y^2-10y=-5$

Calculation

$$\begin{gathered} 3y^2-10y=-5 \\ \Rightarrow \frac{3y^2-10y}{3}=\frac{-5}{3} \\ \Rightarrow y^2-\frac{10}{3}y=\frac{-5}{3} \\ \Rightarrow y^2-\frac{10}{3}y+(\frac{1}{2}\times {\left(\frac{-10}{3}\right))}^2=\frac{-5}{3} \\ \Rightarrow y^2-\frac{10}{3}y+\frac{25}{9}=\frac{-5}{3}+\frac{25}{9} \\ \Rightarrow {(y-\frac{5}{3})}^2=\frac{10}{9} \\ \Rightarrow y-\frac{5}{3}=\pm \sqrt{\frac{10}{9}} \\ \Rightarrow y=\frac{5}{3}\pm \sqrt{\frac{10}{9}} \\ =\frac{5}{3}\pm \frac{\sqrt{10}}{3} \\ =\frac{5+\sqrt{10}}{3},\frac{5-\sqrt{10}}{3} \end{gathered}$$

The required value of$y$is$\frac{5+\sqrt{10}}{3},\frac{5-\sqrt{10}}{3}$