Problem 26 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation analyrically for all complex solutions, giving exact forms in your solution set. Then graph the left side of the equation as \(Y_{1}\) in the suggested viewing window and, using the capabilities of vour calculater, suppert the real solutions. $$\begin{aligned}&-5 x^{3}+13 x^{2}+6 x=0\\\&[-4,4] \text { by }[-2,30]\end{aligned}$$

Step-by-Step Solution

Answer

The complex solutions are \(x = 0, x = -\frac{2}{5}, x = 3\).

Step 1: Identify the Common Factor

First, identify any common factor in the given equation. We have the equation \[-5x^3 + 13x^2 + 6x = 0\].Notice that \(x\) is a common factor in all the terms.

Step 2: Factor Out the Common Factor

Factor out \(x\) from the equation:\[x(-5x^2 + 13x + 6) = 0\].This gives us one solution \(x = 0\). We now need to solve the quadratic equation \(-5x^2 + 13x + 6 = 0\).

Step 3: Use the Quadratic Formula

To solve the quadratic equation, \(-5x^2 + 13x + 6 = 0\), apply the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\].Here, \(a = -5\), \(b = 13\), and \(c = 6\).

Step 4: Calculate the Discriminant

Calculate the discriminant \(b^2 - 4ac\):\[13^2 - 4(-5)(6) = 169 + 120 = 289\].The discriminant is 289, which is a perfect square.

Step 5: Solve Using the Quadratic Formula

Plug the values into the quadratic formula:\[x = \frac{-13 \pm \sqrt{289}}{2(-5)}\].\[\sqrt{289} = 17\].Hence, the solutions are:\[x = \frac{-13 + 17}{-10} = \frac{4}{-10} = -\frac{2}{5}\]and\[x = \frac{-13 - 17}{-10} = \frac{-30}{-10} = 3\].

Step 6: List All Solutions

The complete list of solutions for the equation \(-5x^3 + 13x^2 + 6x = 0\) includes:\[x = 0, -\frac{2}{5}, 3\].These are the exact forms of the solutions.

Step 7: Graph the Equation

Graph the expression \(-5x^3 + 13x^2 + 6x\) within the viewing window \([-4, 4]\) by \([-2, 30]\) on a graphing calculator. Check where it crosses the x-axis to confirm the real solutions \(x = 0\), \(x = -\frac{2}{5}\), and \(x = 3\).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to find the roots of a quadratic equation of the form \(ax^2 + bx + c = 0\). It is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula allows you to solve for \(x\) when you know the coefficients \(a\), \(b\), and \(c\). After plugging the values into the formula, you will either get real or complex solutions.
Real solutions occur when the discriminant, the part under the square root sign \(b^2 - 4ac\), is non-negative, while complex solutions arise when it's negative. Understanding the quadratic formula is essential since it provides a universal method for solving any quadratic equation, even when factoring seems impossible or impractical.

Discriminant

The discriminant is a specific part of the quadratic formula, represented as \(b^2 - 4ac\). It helps determine the nature of the roots of a quadratic equation:

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If the discriminant is zero, the quadratic equation has exactly one real root (also called a repeated root).
• If the discriminant is negative, the quadratic equation has two complex conjugate roots.

In the solution above, we calculated the discriminant as \(13^2 - 4(-5)(6) = 169 + 120 = 289\).
This is a positive perfect square, indicating that the equation has two distinct real roots. By understanding the discriminant, you can quickly assess how many and what type of solutions to expect without conducting full calculations.

Factoring

Factoring is the process of breaking down an expression into a product of simpler expressions, or factors, that when multiplied together give the original expression. In this problem, factoring initially involves recognizing and pulling out any common factors from the entire equation.For example, the given equation \(-5x^3 + 13x^2 + 6x = 0\) shows \(x\) as a common factor across all terms.
Therefore, by factoring out \(x\), the expression simplifies to \(x(-5x^2 + 13x + 6) = 0\). This step is crucial as it makes solving the equation easier. After factoring out the common factor, we can focus on solving the simpler quadratic equation that remains. Factoring is often the first step when solving algebraic equations because it can significantly simplify the problem.

Graphing Calculator

A graphing calculator is a valuable tool in algebra and calculus that helps visualize complex functions and equations. When dealing with equations like \(-5x^3 + 13x^2 + 6x = 0\), a graphing calculator can provide a graphical representation by plotting the function.In the exercise, the equation was graphed within the window \([-4, 4]\) by \([-2, 30]\).
The x-intercepts of the graph correspond to the real solutions of the equation. This means where the graph crosses the x-axis marks solutions where \(y = 0\). Using a graphing calculator can verify solutions found algebraically and provide a visual confirmation of the results. It's especially useful in detecting real solutions and understanding the behavior of complex polynomial expressions.