Chapter 9, Problem 29SGR

To determine

To solve the given equation using graphing.

Answer to Problem 29SGR

The roots of x are -0.8 and 3 .

Explanation of Solution

Given information :

The function provided is $6x^2-13x=15$ . This can be written as $6x^2-13x-15=0$ .

Formula used :

The given equation would be graphed as a function and the points of x-intercepts would be the results. For these points, the x-coordinate shall serve the purpose of the results.

If the roots are not on points that are integers, use the approximation method to compute the roots.

Else, put the x-intercept value in the equation. If the left-hand and right-hand sides of the equations match, it is a success and those will be the values of x.

Calculation :

The graph of the given equation is

  

The zeroes in the graph are at $\left(3,0\right)$ and the other one is not on any integral coordinate. Hence the negative root needs to be computed by approximation.

In this, the parabola one x-intercept is between -1 and 0.

To find the root, make a table of values for x in between these points with intervals on 0.1 . The value of x that is result closest to 0 will be the roots of the equation.

Table of values in between -1 and 0

X	-0.9	-0.8	-0.7	-0.6	-0.5	-0.4	-0.3	-0.2	-0.1
Y	1.56	-0.76	-2.96	-5.04	-7	-8.84	-10.56	-12.16	-13.64

In the tables, the value of y that is closest to 0 is -0.76. Their corresponding x values would be the roots.

Thus, -0.8 is the first root.

For the second root, check if the value of the x-coordinate matches both sides of the given equation.

Put these values in the given equation to recheck.

  $6{\left(3\right)}^2-13\left(3\right)-15=0$

Putting $x=3$ in the given equation.

  $=>54-39-15=0$

Simplifying

  $=>54-54=0$

Hence second value of x is a success.