Question

Solve$3{(b+3)}^2-8=88$

Short answer

The solutions of the quadratic equation:$b=4\sqrt{2}-3,b=-4\sqrt{2}-3$

Step-by-step solution

Step 1. Given information

Consider the quadratic equation

$3{(b+3)}^2-8=88$

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

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

Add 8 from both sides to isolate the binomial term.

localid="1645453235551" $$\begin{gathered} 3{(b+3)}^2=88+8 \\ 3{(b+3)}^2=96 \end{gathered}$$

Divide both sides by 3.

localid="1663772006219" $$\begin{gathered} {(b+3)}^2=\frac{96}{3} \\ =32 \end{gathered}$$

Use the Square Root Property.

$(b+3)=\pm \sqrt{32}$

Step 2. Simplify the radical expression. 

Simplify the radical and solve for $b$.

$$\begin{gathered} b+3=\pm \sqrt{16·2} \\ b=\pm 4\sqrt{2}-3 \end{gathered}$$

Rewrite to show two solutions.

$b=4\sqrt{2}-3,b=-4\sqrt{2}-3$

Step 3. Check the solutions. 

Substitute $b=4\sqrt{2}-3$in the original equation.

role="math" localid="1645453786181" $$\begin{gathered} 3{(b+3)}^2-8=88 \\ 3{(4\sqrt{2}-3+3)}^2-8\stackrel{?}{=}88 \\ 3(16·2)-8\stackrel{?}{=}88 \\ 96-8\stackrel{?}{=}88 \\ 88=88 \end{gathered}$$

Substitute $b=-4\sqrt{2}-3$in the original equation.

$$\begin{gathered} 3{(b+3)}^2-8=88 \\ 3{(-4\sqrt{2}-3+3)}^2-8\stackrel{?}{=}88 \\ 3(16·2)-8\stackrel{?}{=}88 \\ 96-8\stackrel{?}{=}88 \\ 88=88 \end{gathered}$$