Problem 98 - Algebra for College Students

Question

Solve each equation by locating the x-intercepts on a calculator graph. Round approximate answers to two decimal places. $$x^{4}-116 x^{2}+1600=0$$

Step-by-Step Solution

Answer

The x-intercepts are approximately at \(x \approx -10.00, 10.00\).

Step 1: Enter the equation into the calculator

Input the given equation, \(x^{4}-116 x^{2}+1600=0\), into the graphing calculator. Make sure to use the appropriate function input mode.

Step 2: Graph the equation

Plot the graph of the equation on the calculator to visualize the curve. Observe where the graph intersects the x-axis.

Step 3: Identify x-intercepts

Use the calculator’s 'zero' or 'root' function to find the x-intercepts. These are the points where the graph crosses the x-axis (y=0).

Step 4: Approximate the values

Round the x-intercept values to two decimal places as required. Make sure to check multiple intercepts if the graph crosses the x-axis more than once.

Key Concepts

x-intercepts

In solving polynomial equations, finding the x-intercepts is a crucial step. The x-intercepts are the points where the graph of the equation crosses the x-axis, meaning the y-value at these points is zero. For the polynomial equation given \(x^{4}-116x^{2}+1600=0\), the x-intercepts are the solutions to the equation. To find these x-intercepts, we look for values of \(x\) that make the equation equal to zero. This involves graphing the equation and identifying where the curve meets the x-axis.

graphing calculator

A graphing calculator is an essential tool for visualizing polynomial equations. Here's how you can use it effectively:

• First, manually input the equation into the calculator. For example, input \(x^{4}-116 x^{2}+1600=0\) in the appropriate function mode.
• Next, use the graphing feature to plot the equation. The calculator will display a graph where you can see the curve of the polynomial.
• Looking at the graph, focus on where the curve intersects the x-axis. These points give you the x-intercepts, which are the solutions to the polynomial equation.

By effectively using a graphing calculator, you can make the process of finding x-intercepts much simpler and more visual.

polynomial roots

Polynomial roots are another way of referring to the solutions or x-intercepts of a polynomial equation. When we solve \(x^{4}-116 x^{2}+1600=0\), we aim to find these roots. Using the graphing calculator's 'zero' or 'root' function, you can pinpoint the precise values where the polynomial touches the x-axis. These values are:

• 4 real roots (if the polynomial intersects the x-axis at four different points)
• Repeated roots (if the polynomial touches the x-axis but does not cross it)

Understanding polynomial roots helps in comprehending the fundamental nature of polynomial equations and how their solutions relate to the graph.

rounding approximate values

Rounding approximate values is a common requirement in solving polynomial equations. When we use a graphing calculator to find x-intercepts, the results can often be decimal values that are not exact. It is standard practice to round these values to a specified number of decimal places. In the given problem, you need to round to two decimal places. For example:

• If an intercept is calculated as 3.141592, it should be rounded to 3.14.
• Be consistent and round all x-intercepts to ensure uniformity in answers.

Always check if multiple x-intercepts require rounding and adjust each value accordingly.