Problem 41 - A Graphical Approach to Precalculus with Limits

Question

Use an analytic method to solve each equation in part (a). Support the solution with a graph. Then use the graph to solve the inequalities in parts (b) and (c). (a) \(\sqrt[3]{x^{2}-2 x}=\sqrt[3]{x}\) (b) \(\sqrt[3]{x^{2}-2 x}>\sqrt[3]{x}\) (c) \(\sqrt[3]{x^{2}-2 x}<\sqrt[3]{x}\)

Step-by-Step Solution

Answer

(a) Solutions are \(x = 0\) and \(x = 3\). (b) \(-\infty < x < 0\) or \(x > 3\). (c) \(0 < x < 3\).

Step 1: Simplify Equation

First, we will simplify the equation \( \sqrt[3]{x^{2} - 2x} = \sqrt[3]{x} \) by eliminating the cube roots. Cube both sides: \((x^{2} - 2x) = x \).

Step 2: Set Up a Quadratic Equation

The equation \((x^{2} - 2x) = x \) can be rewritten as \( x^{2} - 3x = 0 \).

Step 3: Solve the Quadratic Equation

Factor out an \(x\): \( x(x - 3) = 0 \). This gives the solutions \(x = 0\) and \(x = 3\).

Step 4: Check the Solutions in the Original Equation

Substitute \(x = 0\) and \(x = 3\) back into the original equation \(\sqrt[3]{x^{2} - 2x} = \sqrt[3]{x}\), both solutions satisfy the equation, confirming they are correct.

Step 5: Graph the Functions

Graph the functions \( y = \sqrt[3]{x^{2} - 2x} \) and \( y = \sqrt[3]{x} \) on the same coordinate plane to identify where one is greater or less than the other.

Step 6: Solve the Inequality for (b)

For \(\sqrt[3]{x^{2} - 2x} > \sqrt[3]{x}\), use the graph. The solution is between the x-values for which the curve of \(y = \sqrt[3]{x^{2} - 2x}\) is above \(y = \sqrt[3]{x}\). From the graph, this occurs in the interval \(-\infty, 0 \) and \(3, \infty\).

Step 7: Solve the Inequality for (c)

For \(\sqrt[3]{x^{2} - 2x} < \sqrt[3]{x}\), use the graph. The solution is between \(0\) and \(3\), where \(y = \sqrt[3]{x}\) is above \(y = \sqrt[3]{x^{2} - 2x}\). The interval is \(0, 3\).

Key Concepts

Quadratic Equation

A quadratic equation is an equation that involves a squared term, typically in the form of \( ax^2 + bx + c = 0 \). It is fundamental in algebra, and solving it usually involves finding the values of \( x \) that make the equation true. For instance, the equation \( x^2 - 3x = 0 \) is a quadratic equation we derived from simplifying the original equation.

To solve quadratic equations, you can use various methods like:

• Factoring: This involves rewriting the quadratic as a product of its factors. For example, \( x(x - 3) = 0 \) is the product form of our quadratic equation.
• Using the Quadratic Formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). This powerful formula works for any quadratic equation, but it's often used when the equation cannot be factored easily.
• Completing the Square: This method involves creating a perfect square trinomial on one side of the equation.

In our context, after simplifying the terms, we were able to factor the equation and find the solutions \( x = 0 \) and \( x = 3 \). These solutions reveal the x-intercepts of the parabola, offering insight into the function’s behavior, which becomes essential when performing a graphical analysis.

Graphical Analysis

Graphical analysis helps visualize and better comprehend the relationships and solutions of equations. By graphing the functions that are part of the original problem, you can identify where one function is greater than, less than, or intersects another function. For our exercise, we graphed the functions \( y = \sqrt[3]{x^2 - 2x} \) and \( y = \sqrt[3]{x} \).

When graphing:

• Identify the key points where the functions intersect. This can help confirm solutions from algebraic methods like solving a quadratic equation.
• Observe the intervals where one graph is above or below the other. This is useful in solving inequalities.

In our example, the graphs intersect at \( x = 0 \) and \( x = 3 \), where the values of \( y \) for both functions are equal. By examining the graph, you can observe where \( y = \sqrt[3]{x^2 - 2x} \) is above or below \( y = \sqrt[3]{x} \), aiding in solving inequalities later on.

Inequality Solving

Solving inequalities involving functions can be easier when backed up with a graph. Inequalities show where one function’s value is greater or less than another. In our problem, we considered the inequalities \( \sqrt[3]{x^2 - 2x} > \sqrt[3]{x} \) and \( \sqrt[3]{x^2 - 2x} < \sqrt[3]{x} \).

With graphs:

• Identify the regions where one function is higher or lower than the other. Relate these regions to your inequality questions.
• Use x-values at intersections, where values are equal, to determine boundary points for inequalities.

For \( \sqrt[3]{x^2 - 2x} > \sqrt[3]{x} \), the graph shows that \( y \) is above the other function in the intervals \((-limits\infty, 0) \) and \((3, limits\infty) \). For \( \sqrt[3]{x^2 - 2x} < \sqrt[3]{x} \), \( y \) is lower in the interval \((0, 3) \). Thus, graphical analysis is indispensable in not only visualizing but also accurately defining solution sets for inequalities.