Question

Solve$2{(b+2)}^2=80$

Short answer

The solutions are$b=2\sqrt{10}-2,b=-2\sqrt{10}-2$

Step-by-step solution

Step 1. Given information

Consider the quadratic equation

$2{(b+2)}^2=80$

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

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

Divide both sides by the coefficient 2.

localid="1663771543619" $$\begin{gathered} {(b+2)}^2=\frac{80}{2} \\ =40 \end{gathered}$$

Use the square root property on the binomial.

$b+2=\pm \sqrt{40}$

Step 2. Simplify the radical expression. 

Simplify the radical and solve for $b$.

$$\begin{gathered} b=\pm \sqrt{4·10}-2 \\ b=\pm 2\sqrt{10}-2 \end{gathered}$$

Rewrite to show two solutions.

$b=2\sqrt{10}-2,b=-2\sqrt{10}-2$

Step 3. Check the solutions:   

Substitute $b=2\sqrt{10}-2$in the original equation.

role="math" localid="1645377113748" $$\begin{gathered} 2{(b+2)}^2=80 \\ 2{(2\sqrt{10}-2+2)}^2\stackrel{?}{=}80 \\ 2\left(40\right)\stackrel{?}{=}80 \\ 80=80 \end{gathered}$$

Substitute $b=-2\sqrt{10}-2$in the original equation.

$$\begin{gathered} 2{(b+2)}^2=80 \\ 2{(-2\sqrt{10}-2+2)}^2\stackrel{?}{=}80 \\ 2\left(40\right)\stackrel{?}{=}80 \\ 80=80 \end{gathered}$$