Problem 29 - A Graphical Approach to College Algebra

Question

Solve each equation analytically 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 your calculator, support the real solutions. $$\begin{aligned} &-3 x^{3}-x^{2}+6 x=0\\\ &[-4,4] \text { by }[-10,10] \end{aligned}$$

Step-by-Step Solution

Answer

The exact solutions are \(x = 0\), \(x = \frac{1 + \sqrt{73}}{-6}\), \(x = \frac{1 - \sqrt{73}}{-6}\).

Step 1: Factor the Equation

Start by factoring the given equation \(-3x^3 - x^2 + 6x = 0\). Factor out the greatest common factor, which is \(x\).The equation becomes:\[x(-3x^2 - x + 6) = 0\]

Step 2: Solve for the Zero Product Property

Use the zero product property to solve for \(x\). The solutions to the equation \(x(-3x^2 - x + 6) = 0\) occur when either \(x = 0\) or \(-3x^2 - x + 6 = 0\). So, one solution is \(x = 0\).

Step 3: Solve the Quadratic Equation

Now solve the quadratic equation \(-3x^2 - x + 6 = 0\).Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = -3\), \(b = -1\), and \(c = 6\).Plug into the quadratic formula:\[x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(-3)(6)}}{2(-3)}\]Simplify:\[x = \frac{1 \pm \sqrt{1 + 72}}{-6}\]This gives:\[x = \frac{1 \pm \sqrt{73}}{-6}\].

Step 4: Determine Complex and Real Solutions

The solutions to the equation are: \(x = 0\), \(x = \frac{1 + \sqrt{73}}{-6}\), and \(x = \frac{1 - \sqrt{73}}{-6}\).In exact form, the solutions from the quadratic are real as both expressions contain a square root of a positive number, providing real solutions.

Step 5: Verify the Real Solutions Graphically

Graph the equation \(-3x^3 - x^2 + 6x\) using the viewing window \([-4, 4]\) by \([-10, 10]\).Observe where the graph intersects the x-axis, confirming the real solutions found analytically.The graph should intersect at approximately \(x = 0\), \(x \approx -1.60\), and \(x \approx 2.27\) which are the approximations of the solutions.

Key Concepts

Factoring Equations

Factoring is a fundamental step in solving polynomial equations. When faced with a polynomial, our primary goal is to express it in a product form, making it easier to find its roots. In the given equation \(-3x^3 - x^2 + 6x = 0\), we start by factoring out the greatest common factor (GCF), which is \(x\). This breaks down the equation to \(x(-3x^2 - x + 6) = 0\).
This factorization process has simplified a cubic equation into a product of a linear factor, \(x\), and a quadratic factor, \(-3x^2 - x + 6\). Now, according to the zero product property:

• If \(A \cdot B = 0\), then either \(A = 0\) or \(B = 0\).

Applying this property means we can solve both \(x = 0\) and \(-3x^2 - x + 6 = 0\) separately to find the roots of the original polynomial.
The factorization step is crucial as it sets the stage for easier solution finding and helps visualize the roots of the polynomial!

Quadratic Formula Application

After factoring the polynomial and isolating \(-3x^2 - x + 6 = 0\), we need to solve this quadratic equation. When factoring isn't explicitly possible, the quadratic formula is our go-to tool. The quadratic formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula helps us find the roots by plugging in our specific coefficients: \(a = -3\), \(b = -1\), and \(c = 6\). Replacing these in the formula gives us:\[x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(-3)(6)}}{2(-3)}\]After simplification, we arrive at:\[x = \frac{1 \pm \sqrt{73}}{-6}\]
This result provides two roots, using the "\(\pm\)" symbol to denote both the addition and subtraction versions. This step demonstrates the direct application of a systematic formula to obtain solutions when direct factoring seems challenging or impractical.

Real and Complex Roots

When solving equations such as the ones we've encountered, it is essential to distinguish between real and complex roots. In this particular equation, we obtained the roots:\(x = 0\), \(x = \frac{1 + \sqrt{73}}{-6}\), and \(x = \frac{1 - \sqrt{73}}{-6}\). Thankfully, all roots here are real.
The terms \(\sqrt{73}\) suggest real numbers since 73 is positive. If we were to have a negative number under the square root, our solutions would involve complex numbers, typically in the format of \(a + bi\), where \(i\) is the imaginary unit.
Understanding whether roots are real or complex is crucial. Real roots correspond to graph intersections on the x-axis, while complex roots indicate a different behavior, often parallel translation in graphs.

Graphical Verification

Verifying analytical solutions with a graph provides an additional layer of confidence. Using graphing tools, we graph \(-3x^3 - x^2 + 6x\) and inspect it within the window \([-4, 4]\) by \([-10, 10]\).
The graph complements our analytical results as it shows where the curve intersects the x-axis. The intersections at \(x = 0\), \(x \approx -1.60\), and \(x \approx 2.27\) should be visible, aligning with our real roots.

• \(x = 0\) directly matches our linear factor's root.
• \(x \approx -1.60\) and \(x \approx 2.27\) are approximations of the remaining real solutions derived from the quadratic formula.

Graphical verification is an excellent way to visualize the roots, ensuring our calculations are accurate and valid. It bridges the gap between analytical solutions and visual interpretation.