Question

Solve$5r^2-2=34$

Short answer

The solutions of the quadratic equation:$r=\frac{6\sqrt{5}}{5},r=-\frac{6\sqrt{5}}{5}$

Step-by-step solution

Step 1. Given information

Consider the quadratic equation

$5r^2-2=34$

The objective is to solve the quadratic equation by using square root property.

If$x^2=k$, then$x=\pm \sqrt{k}$

Isolate the quadratic term.

$5r^2=36$

Divide by 5 to make the coefficient 1.

$$\begin{gathered} \frac{5r^2}{5}=\frac{36}{5} \\ {r}^2=\frac{36}{5} \end{gathered}$$

Use the Square Root Property.

localid="1645369492576" $r=\pm \sqrt{\frac{36}{5}}$

Step 2. Simplify the radical expression.

Rewrite the radical as a fraction of square roots.

$r=\pm \frac{\sqrt{36}}{\sqrt{5}}$

Rationalize the denominator.

$$\begin{gathered} r=\pm \frac{\sqrt{36}}{\sqrt{5}}·\frac{\sqrt{5}}{\sqrt{5}} \\ r=\pm \frac{6\sqrt{5}}{5} \end{gathered}$$

Rewrite to show two solutions.

$r=\frac{6\sqrt{5}}{5},r=-\frac{6\sqrt{5}}{5}$

Step 3. Check the solutions:

Substitute $r=\frac{6\sqrt{5}}{5}$in the original equation.

$$\begin{gathered} 5r^2-2=34 \\ 5{\left(\frac{6\sqrt{5}}{5}\right)}^2-2\stackrel{?}{=}34 \\ 5\left(\frac{36·5}{25}\right)-2\stackrel{?}{=}34 \\ 36-2\stackrel{?}{=}34 \\ 34=34 \end{gathered}$$

Substitute $x=-\frac{6\sqrt{5}}{5}$in the original equation.

$$\begin{gathered} 5r^2-2=34 \\ 5{\left(-\frac{6\sqrt{5}}{5}\right)}^2-2\stackrel{?}{=}34 \\ 5\left(\frac{36·5}{25}\right)-2\stackrel{?}{=}34 \\ 36-2\stackrel{?}{=}34 \\ 34=34 \end{gathered}$$