Question

Find all real solutions of the equation, rounded to two decimals.

$x^3-2x^2-x-1=0$

Short answer

The real solution of the equation$x^3-2x^2-x-1=0$ is $x=2.55$.

Step-by-step solution

Step 1. Apply the concept of intercepts.

The x-intercepts are the points on x-axis where the graph of the equation intersects the x-axis.

The y-intercepts are the points on y-axis where the graph of the equation intersects the y-axis.

The real solution of equation$y=f\left(x\right)$ are the points where the graph intersects the x-axis.

Step 2. Graph the equation.

Open the TI-83 calculator and follow the steps below.

Step 1: Press$\text{Y}=$key.

Step 2: Enter the function ${\text{Y}}_1=x^3-2x^2-x-1$.

Step 3: Press the WINDOW key.

Step 4: Make the following changes.

$\begin{array}{c}X\min =-5\\ X\max =5\\ Y\min =-5\\ Y\max =5\end{array}$

Step 5: Press the TRACE key.

The result obtained on screen is the graph of the equation $x^3-2x^2-x-1=0$.

There is only one real solution to equation as graph intersects the x-axis only once.

Step 3. Find the x-intercept.

Once, the graph is obtained on screen, follow the steps below to compute the x-intercept.

Step 1: Press 2nd key and then TRACE key.

Step 2: Select the second option “zero”.

Step 3: To enter the left bound, move the curve to the left of the x-intercept and press ENTER key.

Step 4: To enter the right bound, move the curve to the left of the x-intercept and press ENTER key.

Step 5: For a guess value, press ENTER key.

The result obtained on screen is provided below.

Therefore, x-intercept is 2.5468.

Round it off to two decimal places.

Therefore, x-intercept is 2.55 and real solution is $x=2.55$.

Thus, the real solution of the equation$x^3-2x^2-x-1=0$ is $x=2.55$.