Problem 30 - 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. $$\begin{aligned}&-4 x^{3}-x^{2}+4 x=0\\\&[-4,4] \text { by }[-10,10]\end{aligned}$$

Step-by-Step Solution

Answer

Solutions: \(x = 0, \frac{1+\sqrt{65}}{-8}, \frac{1-\sqrt{65}}{-8}\); real solutions confirmed by graph.

Step 1: Factor the Equation

First, factor the left side of the equation \(-4x^3 - x^2 + 4x = 0\).Start by factoring out the common factor of \(x\):\[x(-4x^2 - x + 4) = 0\]Now, solve \(-4x^2 - x + 4 = 0\) separately.

Step 2: Apply the Quadratic Formula

To solve the quadratic equation \(-4x^2 - x + 4 = 0\), we use the quadratic formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here, \(a = -4\), \(b = -1\), and \(c = 4\). Plug these into the formula:\[x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(-4)(4)}}{2(-4)}\]Simplifying further gives:\[x = \frac{1 \pm \sqrt{1 + 64}}{-8}\]\[x = \frac{1 \pm \sqrt{65}}{-8}\]

Step 3: Solve the Linear Factor

The factored equation from Step 1 gives us the linear factor \(x = 0\). Therefore, the equation has at least one solution where \(x = 0\).

Step 4: Identify All Solutions

The complete set of solutions for the equation includes the linear solution from Step 3 and the solutions from the quadratic solution in Step 2:\[x = 0\]\[x = \frac{1 + \sqrt{65}}{-8}\]\[x = \frac{1 - \sqrt{65}}{-8}\]Thus, the exact complex solutions to the equation are \(x = 0, \frac{1 + \sqrt{65}}{-8}, \) and \( \frac{1 - \sqrt{65}}{-8}\).

Step 5: Graph and Confirm Real Solutions

Using a calculator, graph \(-4x^3 - x^2 + 4x\) over the viewing window \([-4, 4]\) by \([-10, 10]\). Observe where the graph crosses the x-axis to confirm real solutions.The real solutions are confirmed to be \(x = 0\) and the approximate values of the quadratic roots. Verify the solutions by checking the graph.

Key Concepts

Factoring Equations

Factoring equations is a key skill in algebra that helps to break down complex expressions into simpler components. In the context of solving polynomial equations, factoring allows us to express the equation as a product of simpler expressions, which can then be solved independently.
In our exercise, we tackled the polynomial \(-4x^3 - x^2 + 4x = 0\). The first step was to identify a common factor in each term, which is \(x\). Factoring out \(x\) gives us:

• \(x(-4x^2 - x + 4) = 0\)

By factoring, we reduced a cubic equation to a simpler problem: a linear factor \(x\) and a quadratic equation \(-4x^2 - x + 4 = 0\). Solving each of these separately gives us the roots of the original equation, which includes finding the factors of the quadratic part.

Quadratic Formula

The quadratic formula is an essential tool for finding solutions to quadratic equations. These are equations in the form \(ax^2 + bx + c = 0\). The formula is given by:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

In our exercise, the quadratic part of the equation was \(-4x^2 - x + 4 = 0\) with \(a = -4\), \(b = -1\), and \(c = 4\). Plugging these into the formula, we obtained the solutions:

• \(x = \frac{1 \pm \sqrt{65}}{-8}\)

It's important to simplify under the square root, ensuring we handle any real or complex results as they arise. This method is reliable and often necessary when factoring is not straightforward or when complex solutions exist.

Graphing Equations

Graphing equations allows us to visually interpret the solutions and behavior of the function. Using a graphing calculator, plotting the equation helps to confirm the roots and understand their positions relative to the x-axis.
In this exercise, we graphed \(-4x^3 - x^2 + 4x\) in the window \([-4, 4]\) by \([-10, 10]\). The graph provided a clear picture of where the polynomial crosses the x-axis, showing us the real intersections:
The crossing points on the x-axis at \(x = 0\) and around the approximate roots given by the quadratic formula show the real solutions. Graphs are an invaluable tool for supporting analytic solutions, offering a means to verify and visualize our results.

Real Solutions Verification

Verification of real solutions is a critical step in solving equations to ensure accuracy. Once we have the potential solutions, both from factoring and the quadratic formula, verifying them against the graph is crucial.
In this problem, we identified potential solutions \(x = 0\) and the roots from the quadratic formula as real. By graphing the equation, we checked where the graph crosses the x-axis:
These real intersections confirmed our solutions, showing \(x = 0\) visibly. The approximate intersection points align with the calculation results from the quadratic, ensuring that our analytic methods were correct. Verifying solutions using graphs provides a practical tool to double-check work and understand the nature of the roots.