Chapter 1.1, Problem 51E

Solution

a)

To find: The real solutions of $10x^3+7.5x^2-54.85x+37.95=0$ .

The three real solutions of $10x^3+7.5x^2-54.85x+37.95=0$ are $-3$ , $1.1$ and $1.15$ .

Given Information:

The equation $10x^3+7.5x^2-54.85x+37.95=0$ .

Concept Used:

The real solutions of an equation are the values of x where the graph cuts the x- axis.

Calculation:

The graph of $10x^3+7.5x^2-54.85x+37.95=0$ is shown in figure 1.

  

Figure 1

The graph in figure 2 shows the section of the graph of $10x^3+7.5x^2-54.85x+37.95=0$ in between $1.08\le x\le 1.16$ .

  

Figure 2

Therefore, it follows that the three real solutions of $10x^3+7.5x^2-54.85x+37.95=0$ are $-3$ , $1.1$ and $1.15$ .

b)

To find: The real solutions of $x^3+x^2-4.99x+3.03=0$ .

The real solution of $x^3+x^2-4.99x+3.03=0$ is $-3$ .

Given Information:

The equation $x^3+x^2-4.99x+3.03=0$ .

Concept Used:

The real solutions of an equation are the values of x where the graph cuts the x- axis.

Calculation:

The graph of $x^3+x^2-4.99x+3.03=0$ is shown in figure 3.

  

Figure 3

The graph in figure 4 shows the section of the graph of $x^3+x^2-4.99x+3.03=0$ in between $0.9\le x\le 1.1$ .

  

Figure 4

Therefore, it follows that the real solution of $x^3+x^2-4.99x+3.03=0$ is $-3$ .