Problem 123 - College Algebra

Question

Use a graphing utility and the graph's \(x\) -intercepts to solve each equation. Check by direct substitution. A viewing rectangle is given. $$\begin{aligned} &-x^{4}+4 x^{3}-4 x^{2}-0\\\ &[-6,6,1] \text { by }[-9,2,1] \end{aligned}$$

Step-by-Step Solution

Answer

The roots of the equation, \(-x^{4}+4 x^{3}-4 x^{2}=0\), are the x-intercepts of its graph. Check each x-intercept by substituting it back into the original equation to ensure it satisfies the equation.

Step 1: Plot the function

Using a graphing utility, plot the function \(-x^{4}+4 x^{3}-4 x^{2}\). The given range \([-6,6,1]\) is for the x-values and \([-9,2,1]\) for the y-values.

Step 2: Locate the x-intercepts of the graph

Identify the values of \(x\) where the graph of the function intersects or crosses the x-axis. These are the \(x\)-intercepts.

Step 3: Check solutions by direct substitution

Check each \(x\)-intercept obtained in step 2 by substituting it back into the original polynomial equation. If the resulting value equals 0 for each instance of \(x\), then these are the correct solutions.

Key Concepts

Graphing Utility

A graphing utility is a tool, such as a graphing calculator or software program, that helps to visualize mathematical equations by displaying their graphs. When it comes to solving polynomial equations, a graphing utility can be invaluable. It takes the abstract algebraic formula and turns it into a visual representation, allowing students to see the behavior of the polynomial over a range of values.

For example, the polynomial equation \( -x^4 + 4x^3 - 4x^2 \) can appear daunting at first. When input into a graphing utility and observed over the given range, \( [-6,6,1] \) for the x-axis and \( [-9,2,1] \) for the y-axis, the graph provides a clear picture of where the function rises and falls, and most importantly, where it intersects the x-axis, which are the solutions to the equation.

X-Intercepts

X-intercepts are the points where a graph crosses the x-axis. These points are crucial in solving polynomial equations as they represent the solutions or roots of the equation. Since the graph is a visual representation of all possible values that satisfy the equation, the x-intercepts are those where the value of the polynomial is zero.

Identifying the x-intercepts on a graph can be straightforward: they are where the plot touches or crosses the horizontal axis. In our exercise, locating the x-intercepts of \( -x^4 + 4x^3 - 4x^2 \) would reveal the values of \( x \) for which the equation holds true, which are the answers we seek. Once these intercepts are found using the graphing utility, one can proceed to verify them through direct substitution.

Direct Substitution

Direct substitution is a method of verification used to confirm the solutions obtained by other means—such as by using a graphing utility. The process involves replacing the variable \( x \) in the original polynomial equation with each of the identified x-intercepts. If, after the substitution, the equation equals zero, then that value of \( x \) is indeed a solution to the polynomial equation.

This step ensures that any potential rounding or graphical errors are caught and that the solutions are accurate. For instance, if one of the x-intercepts from the graph of the polynomial \( -x^4 + 4x^3 - 4x^2 \) is 2, we can substitute \( x \) with 2 in the equation. If the result is zero, \( 2 \) is confirmed as a valid solution.