Question

Solve$4t^2+2t=20$by completing the square.

Short answer

The values of $t$for the equation$4t^2+2t=20$by completing the squares are$2,-\frac{5}{2}$.

Step-by-step solution

Step 1. Given Information.

The equation is$4t^2+2t=20$.

Step 2. Using completing the square method.

Given, $4t^2+2t=20$

Dividing both sides by $4$,

$$\begin{gathered} \Rightarrow \frac{4t^2+2t}{4}=\frac{20}{4} \\ \Rightarrow t^2+\frac{1}{2}t=5 \end{gathered}$$

Taking the square of the half of $\frac{1}{2}$and adding it on both sides,

$$\begin{gathered} \Rightarrow t^2+\frac{1}{2}t+{\left(\frac{1}{2}\left(\frac{1}{2}\right)\right)}^2=5+{\left(\frac{1}{2}\left(\frac{1}{2}\right)\right)}^2 \\ \Rightarrow t^2+\frac{1}{2}t+\frac{1}{16}=5+\frac{1}{16} \\ \Rightarrow t^2+\frac{1}{2}t+\frac{1}{16}=\frac{81}{16} \end{gathered}$$

Factoring the right side as a binomial square,

$\Rightarrow {\left(t+\frac{1}{4}\right)}^2=\frac{81}{16}$

Taking the square root on both sides,

$$\begin{gathered} \Rightarrow t+\frac{1}{4}=\sqrt{\frac{81}{16}} \\ \Rightarrow t+\frac{1}{4}=\pm \frac{9}{4} \end{gathered}$$

Solving for $t$,

$$\begin{gathered} \Rightarrow t=-\frac{1}{4}\pm \frac{9}{4} \\ \Rightarrow t=-\frac{1}{4}+\frac{9}{4} \\ or,t=-\frac{1}{4}-\frac{9}{4} \\ \Rightarrow t=2,-\frac{5}{2} \end{gathered}$$