Question

Solve the quadratic equation using square root property.

$\frac{2}{5}{a}^2+3=11$.

Short answer

The solutions are$\pm 2\sqrt{5}$.

Step-by-step solution

Given Information. 

The Given equation is$\frac{2}{5}{a}^2+3=11$.

Isolate the quadratic term and makes its coefficient one.  

$$\begin{gathered} \frac{2}{5}{a}^2=11-3 \\ \frac{2}{5}{a}^2=8 \end{gathered}$$

Divide by $\frac{2}{5}$both sides.

$$\begin{gathered} a^2=8\times \frac{5}{2} \\ {a}^2=\frac{40}{2} \\ {a}^2=20 \end{gathered}$$

 Use the square root property. 

$a=\sqrt{20}$

Simplify the radical by factoring the radicand.

$$\begin{gathered} a=\pm \sqrt{2·2·5} \\ a=\pm 2\sqrt{5} \end{gathered}$$

Check the solutions. 

Substitute $a=2\sqrt{5}$

$$\begin{gathered} \frac{2}{5}(2\sqrt{5}{)}^2+3\stackrel{?}{=}11 \\ \frac{2}{5}(20)+3\stackrel{?}{=}11 \\ 8+3\stackrel{?}{=}11 \\ 11=11True \end{gathered}$$

Substitute $a=-2\sqrt{5}$

$$\begin{gathered} \frac{2}{5}(-2\sqrt{5}{)}^2+3\stackrel{?}{=}11 \\ \frac{2}{5}(20)+3\stackrel{?}{=}11 \\ 8+3\stackrel{?}{=}11 \\ 11=11True \end{gathered}$$