Q.9.11 - Intermediate Algebra

Question

Solve $3{(a-3)}^2=54$

Step-by-Step Solution

Answer

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

Step 1. Given information

Consider the quadratic equation

$3{(a-3)}^2=54$

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

If $x^2=k$, then $x=\pm \sqrt{k}$

Divide both sides by the coefficient 3.

$$\begin{gathered} {(a-3)}^2=\frac{54}{3} {(a-3)}^2 \\ =18 \end{gathered}$$

Use the Square Root Property on the binomial.

 $a-3=\pm \sqrt{18}$

Step 2. Simplify the radical expression.

Simplify the radical and solve for $a$.

$$\begin{gathered} a=\pm \sqrt{9·2}+3 \\ a=\pm 3\sqrt{2}+3 \end{gathered}$$

Rewrite to show two solutions.  

$a=3\sqrt{2}+3, a=-3\sqrt{2}+3$

Step 3. Check the solutions:

Substitute $a=3\sqrt{2}+3$ in the original equation.

$$\begin{gathered} 3{(a-3)}^2=54 \\ 3{(3\sqrt{2}+3-3)}^2\stackrel{?}{=}54 \\ 3{\left(3\sqrt{2}\right)}^2\stackrel{?}{=}54 \\ 3\left(18\right)\stackrel{?}{=}54 \\ 54=54 \end{gathered}$$

Substitute $a=-3\sqrt{2}+3$ in the original equation.

$$\begin{gathered} 3{(a-3)}^2=54 \\ 3{(-3\sqrt{2}+3-3)}^2\stackrel{?}{=}54 \\ 3{\left(3\sqrt{2}\right)}^2\stackrel{?}{=}54 \\ 3\left(18\right)\stackrel{?}{=}54 \\ 54=54 \end{gathered}$$