Q.14 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 5–16, use a graphing utility to approximate the real solutions, if any, of each equation rounded to two decimal places. All solutions lie between - 10 and 10.$\frac{1}{4}{x}^3 - 5x = \frac{1}{5}{x}^2- 4$

Step-by-Step Solution

Answer

The approximate values of X are $x=0.8,\hspace{0.17em}x=-4.47213\dots ,\hspace{0.17em}x=4.47213\dots$

Step 1. Given equation and simplification

$$\begin{gathered} \frac{1}{4}{x}^3-5x=\frac{1}{5}{x}^2-4 \\ \Rightarrow 5x^3-100x=4x^2-80 \\ \Rightarrow 5x^3-4x^2-100x+80=0 \end{gathered}$$

Step 2.Graph and x-intercepts

X-intercepts are $x=\frac{4}{5},\hspace{0.17em}x=-2\sqrt{5},\hspace{0.17em}x=2\sqrt{5}$