Chapter 4.2, Problem 45PPS

To determine

To solve: the given equation. If exact roots cannot be found, state the consecutive integers between which the roots are located.

Answer to Problem 45PPS

  $-1\text{and}0$ and $4\text{and}5$ .

Explanation of Solution

Given:

The given equation is $-3x^2+11x+9=1$ .

Calculation:

The given equation is expressed as

  $-3x^2+11x+8=0\left\{\text{subtract}1\text{ on both the sides}\right\}$ .

Now, graph the related quadratic function to show that x -intercept of the graph. Therefore,

  

Looking at the graph, the parabola crosses the x -axis, therefore, there is two real solution for the graph.

The axis of symmetric is

  $\begin{array}{l}x=\frac{-b}{2a}\left\{\text{equation for symmetry}\right\}\\ x=\frac{-\left(11\right)}{2\left(-3\right)}\{\text{substitutes these value}}\\ x=\frac{11}{6}\end{array}$

Now write the table for the given function.

$x$	$-1$	$0$	$\frac{11}{6}$	$4$	$5$
$f\left(x\right)$	$-6$	$-8$	$\frac{217}{12}$	$4$	$-12$

Based on the table and the graph, the x -intercept of the graph indicate that one solution is between $-1\text{and}0$ and the other solution is between $4\text{and}5$ .

Hence, the solution of the given equation is between $-1\text{and}0$ and $4\text{and}5$ .