Chapter 5.6, Problem 4AYU

Approximate the solution(s) to $x^3-2x+2\text{ = 0}$ using a graphing utility. (pp. 26-28)

To determine

To find: Approximate the solution(s) to $x^3-2x+2\text{ = 0}$ using a graphing utility.

Answer to Problem 4AYU

The solution is the $x\text{-}\text{value}$ of the point of intersection $x\hspace{0.17em}=\hspace{0.17em}-1.769292\hspace{0.17em}=\hspace{0.17em}-1.77$.

Explanation of Solution

Given:

$x^3 - 2x + 2\text{ = 0}$

Calculation:

The ZERO (or ROOT) feature of a graphing utility can be used to find the solutions of an equation when one side of the equation is 0. In using this feature to solve equations, make use of the fact that when the graph of an equation in two variables, $x$ and $y$, crosses or touches the $x\text{-}\text{axis}$ then $y\text{=}\text{0}$. For this reason, any value of $x$ for which $y\text{=}\text{0}$ will be a solution to the equation. That is, solving an equation for $\text{x}$ when one side of the equation is 0 is equivalent to finding where the graph of the corresponding equation in two variables crosses or touches the $x\text{-}\text{axis}$.

When a graphing utility is used to solve an equation, usually approximate solutions are obtained. Unless otherwise stated, we shall follow the practice of giving approximate solutions as decimals rounded to two decimal places.

Steps for Approximating Solutions of Equations Using ZERO (or ROOT).

Step 1: Write the equation in the form $x^3-2x+2=0$.

Step 2: Graph $Y1=x^3-2x+2$.

Be sure that the graph is complete. That is, be sure that all the $x\text{-}\text{intercepts}$ are shown on the screen.

Step 3: Use ZERO (or ROOT) to determine each $x\text{-}\text{intercept}$ of the graph.

See on the graph, we can find the intersection of $x\text{-}\text{coordinate}$ values and solving the graph algebraically.

We get the solution of the $x\text{-}\text{value}$ of the point of intersection $x=-1.769292=-1.77$.