Question

Solve$\frac{3}{4}{y}^2-3=18$

Short answer

The solutions of the quadratic equation:$y=2\sqrt{7},y=-2\sqrt{7}$

Step-by-step solution

Step 1. Given information

Consider the quadratic equation

$\frac{3}{4}{y}^2-3=18$

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

If$x^2=k$, then$\frac{3}{4}{y}^2-3=18$

Isolate the quadratic term.

$$\begin{gathered} \frac{3}{4}{y}^2=18+3 \\ \frac{3}{4}{y}^2=21 \end{gathered}$$

Multiply by $\frac{4}{3}$to make the coefficient 1.

localid="1645365650920" $$\begin{gathered} \frac{4}{3}·\frac{3}{4}{y}^2=\frac{4}{3}·21 \\ {y}^2=28 \end{gathered}$$

Use the Square Root Property.

localid="1645365679482" $y=\pm \sqrt{28}$

Step 2. Simplify the expression.   

Simplify the radical.

$$\begin{gathered} y=\pm \sqrt{4·7} \\ y=\pm 2\sqrt{7} \end{gathered}$$

Rewrite to show two solutions.

$y=2\sqrt{7},y=-2\sqrt{7}$

Step 3. Check the solutions:   

Substitute $y=2\sqrt{7}$in the original equation.

$$\begin{gathered} \frac{3}{4}{y}^2-3=18 \\ \frac{3}{4}{\left(2\sqrt{7}\right)}^2-3\stackrel{?}{=}18 \\ \frac{3}{4}\left(28\right)-3\stackrel{?}{=}18 \\ 18=18 \end{gathered}$$

And substitute role="math" localid="1645366022939" $y=-2\sqrt{7}$in the original equation.

$$\begin{gathered} \frac{3}{4}{y}^2-3=18 \\ \frac{3}{4}{(-2\sqrt{7})}^2-3\stackrel{?}{=}18 \\ \frac{3}{4}\left(28\right)-3\stackrel{?}{=}18 \\ 18=18 \end{gathered}$$