Question

Solve the equation both algebraically and graphically. (Lesson 1.11)

$6{\left(x+2\right)}^5=64$

Short answer

The solution by algebraically is no real solutions $x=-0.39$and graphically is given below:

Step-by-step solution

Step 1. Definition of the intersection point.

An intersection is a single point where two lines meet or cross each other.

Step 2. Solve equation algebraically.

We have$6{\left(x+2\right)}^5=64$

Divide both sides by 64:

$\begin{array}{l}\frac{6{\left(x+2\right)}^5}{6}=\frac{64}{6}\\ {\left(x+2\right)}^5=\frac{32}{3}\end{array}$

Take the fifth root to both sides:

$\begin{array}{l}\sqrt[5]{{\left(x+2\right)}^5}=\sqrt[5]{\frac{32}{3}}\\ \left(x+2\right)=\sqrt[5]{\frac{32}{3}}\\ x=\sqrt[5]{\frac{32}{3}}-2\\ x\approx -0.39\end{array}$

Step 3. Solve the equation graphically.

$6{\left(x+2\right)}^5=64$

Now, equate the left side of the equation to $y_1$and the right side to $y_2$.

$\begin{array}{l}{y}_1=6{\left(x+2\right)}^5\\ y_2=64\end{array}$

Draw the graph:

As we can see that both lines are intersecting at $\left(-0.39,0\right)$.

So, the solution is$x=-0.39$