Chapter 2.3, Problem 46E

To determine

To find: The real number that is exactly $2$ more than its cube.

Answer to Problem 46E

The real number that is exactly $2$ more than its cube are $x=-1.521$.

Explanation of Solution

Given information: The number is exactly $2$ more than its cube.

Calculation:

Assume that $x$ be the real number such that the number is exactly $2$ more than its cube. So, the difference between the number and cube of number is $2$ . So,

  $x-x^3=2$

Solve the equation $x-x^3=2$ with the help of graph.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter left hand side of the equation $x-x^3=2$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{\text{X}}\boxed{-}\boxed{\text{X}}\boxed{^}\boxed{3}$ . Now press the $\boxed{\text{ENTER}}$ key and enter right hand side of the equation $x-x^3=2$ after the symbol ${\text{Y}}_2$ . Enter the keystrokes $\boxed{2}$.

The display will show the equations,

  $\begin{array}{l}{\text{Y}}_{\text{1}}=\text{X}-\text{X}^3\\ {\text{Y}}_2=2\end{array}$

Now, press the $\boxed{\text{GRAPH}}$ key. To find the intersection point, enter the keystrokes $\boxed{2\text{nd}}\boxed{\text{TRACE}}\boxed{5}\boxed{\text{ENTER}}\boxed{\text{ENTER}}\boxed{\text{ENTER}}$ key to produce the graph of given function in standard window as shown below.

  

Figure (1)

As observed from the graph, both the graphs intersect at only one point $x=-1.521$.

So, the solution of the equation $x-x^3=2$ is $x=-1.521$.

Therefore, the real number that is exactly $2$ more than its cube are $x=-1.521$.