Chapter 11.5, Problem 41PPE

a.

To determine

To write two functions using the expressions $\frac{6}{x^2}+1=\frac{{\left(x+7\right)}^2}{6}$ .

Answer to Problem 41PPE

  $\begin{array}{l}{y}_1=\frac{6}{x^2}+1,\\ y_2=\frac{{\left(x+7\right)}^2}{6}\end{array}$

Explanation of Solution

Given information:

The expression $\frac{6}{x^2}+1=\frac{{\left(x+7\right)}^2}{6}$ .

Calculation:

The two functions for the given expression are

  $\begin{array}{l}{y}_1=\frac{6}{x^2}+1,\\ y_2=\frac{{\left(x+7\right)}^2}{6}\end{array}$

The graph of the functions is shown below

  

b.

To determine

The coordinate of the points of intersection.

Answer to Problem 41PPE

  $\left(-9.529,1.066\right),\left(-4.157,1.347\right),\left(-1.123,5.756\right),\text{ and }\left(0.809,10.164\right)$

Explanation of Solution

Given information:

The expression $\frac{6}{x^2}+1=\frac{{\left(x+7\right)}^2}{6}$ .

Calculation:

From the above graph it is clear that the coordinates of the intersection points are $\left(-9.529,1.066\right),\left(-4.157,1.347\right),\left(-1.123,5.756\right),\text{ and }\left(0.809,10.164\right)$

c.

To determine

Are the x -coordinates of the points of intersection solutions of the equation?

Answer to Problem 41PPE

  $\text{Yes}$

Explanation of Solution

Calculation:

Since the graphs represent the left hand side and right hand side of the given equation, so the intersection points represent the points where the left hand side is equal to the right hand side expression.

So, yes the x -coordinates of the points of intersection represent the solution of the equation.