Chapter 4.5, Problem 60E

To determine

To solve: the equation $f\left(x\right)=g\left(x\right)$ for the given function graphically and algebraically.

Answer to Problem 60E

  $x=2,4$

Explanation of Solution

Given information:

Given functions are,

  $f\left(x\right)=x^4-5x^3+2x^2+8x$

And

  $g\left(x\right)=-x^2+6x-8$

Calculation:

Consider the given functions.

  $f\left(x\right)=x^4-5x^3+2x^2+8x$ …… (1)

  $g\left(x\right)=-x^2+6x-8$ …… (2)

Now, the points of intersection of the graph of $f\left(x\right)$ and $g\left(x\right)$ will be the solution of $f\left(x\right)=g\left(x\right)$ .

Construct the graphs of $f\left(x\right)$ and $g\left(x\right)$ in same set of axes.

  

Here, the point of intersection are $\left(2,0\right)$ and $\left(4,0\right)$ .

Thus, the solutions are $x=2,4$ .

Now, solve $f\left(x\right)=g\left(x\right)$ algebraically.

  $\begin{array}{c}{x}^4-5x^3+2x^2+8x=-x^2+6x-8\\ x^4-5x^3+2x^2+8x+x^2-6x+8=0\\ x^4-5x^3+3x^2+2x+8=0\\ \left(x-2\right)\left(x-4\right)\left(x^2+x+1\right)=0\end{array}$

Now, $x^2+x+1\ne 0$ for any real x .

So, $\begin{array}{l}x-2=0\\ x-4=0\end{array}$

Hence, $x=2,4$ are the only real solution of $f\left(x\right)=g\left(x\right)$ .