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

Step-by-Step Solution

Answer

Solutions: \(x = 0, \frac{1+\sqrt{65}}{-8}, \frac{1-\sqrt{65}}{-8}\). Graph confirms real solutions.

Step 1: Factor the Equation

The given equation is \(-4x^3 - x^2 + 4x = 0\). First, factor out the common factor from all terms:\(x(-4x^2 - x + 4) = 0\). This gives us the first solution: \(x = 0\).

Step 2: Solve the Quadratic

The quadratic part of the equation is \(-4x^2 - x + 4 = 0\). We solve this using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = -4\), \(b = -1\), \(c = 4\).

Step 3: Calculate the Discriminant

Compute the discriminant \(\Delta = b^2 - 4ac\):\(\Delta = (-1)^2 - 4(-4)(4) = 1 + 64 = 65\). Since the discriminant is positive, there are two real solutions.

Step 4: Apply the Quadratic Formula

Now plug the values into the quadratic formula:\(x = \frac{-(-1) \pm \sqrt{65}}{2(-4)}\). Simplify this to get:\(x = \frac{1 \pm \sqrt{65}}{-8}\). This results in two solutions:\(x = \frac{1 + \sqrt{65}}{-8}\) and \(x = \frac{1 - \sqrt{65}}{-8}\).

Step 5: Write the Solution Set

Combine all solutions into a solution set:\[ x = 0, \frac{1 + \sqrt{65}}{-8}, \frac{1 - \sqrt{65}}{-8} \]. These represent all the complex numbers (real in this case) that satisfy the equation.

Step 6: Graph the Left Side of the Equation

Graph \(y_1 = -4x^3 - x^2 + 4x\) using a graphing calculator or software in the window \([-4,4]\) by \([-10,10]\). Check intersections with the x-axis, which should occur at the real solutions derived.

Step 7: Verify Real Solutions with Calculator

Using the calculator's graphing capabilities, confirm that all x-intercepts (real solutions) lie within the derived solutions. Ensure the parabola correctly intersects the x-axis at \(x = 0\) and the approximate decimals of \(x = \frac{1 + \sqrt{65}}{-8}\) and \(x = \frac{1 - \sqrt{65}}{-8}\).

Key Concepts

Factoring Polynomials

Factoring is a process used to simplify polynomials by expressing them as a product of simpler polynomials. To factor the equation \[-4x^3 - x^2 + 4x = 0\], we begin by identifying any common factors in each term. In this case, each term includes the variable \(x\), which can be factored out to produce:

• \(x(-4x^2 - x + 4) = 0\)

Here, \(x\) is the common factor. When a factor expression is equal to zero, any factor itself can be zero. Therefore, we immediately identify \(x = 0\) as one of our solutions.

The polynomial \(-4x^2 - x + 4\) now remains, and to fully solve the equation, we must solve this quadratic component separately. Factoring not only simplifies polynomials but also helps in finding solutions, whether they are real or complex numbers, by breaking down complex expressions into manageable pieces.

Quadratic Formula

The quadratic formula is a universal method to find solutions to any quadratic equation of the form \(ax^2 + bx + c = 0\). For our equation, the quadratic formula helps to solve the quadratic polynomial \(-4x^2 - x + 4 = 0\). The formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]where \(a = -4\), \(b = -1\), and \(c = 4\).

• First, calculate the discriminant \(\Delta = b^2 - 4ac\):\(\Delta = 1 + 64 = 65\).
• Since \(\Delta > 0\), it indicates two distinct real solutions.

Substitute these values into the quadratic formula, yielding:\[x = \frac{1 \pm \sqrt{65}}{-8}\]Simplifying offers two real solutions: \(x = \frac{1 + \sqrt{65}}{-8}\) and \(x = \frac{1 - \sqrt{65}}{-8}\).
The quadratic formula provides us a reliable way to find the roots of any quadratic equation, and is especially useful when simple factoring isn't possible.

Graphing Polynomials

Graphing polynomials gives a visual representation of the solutions and behavior of polynomial equations. For the equation \(y_1 = -4x^3 - x^2 + 4x\), graphing it helps us understand where the polynomial intersects the x-axis, which corresponds to the solutions of the equation.
Here’s how you can approach graphing this polynomial:

• Use graphing software or a calculator to input the equation and set the viewing window from \([-4,4]\) across the x-axis and \([-10,10]\) on the y-axis.
• Look for the x-intercepts, which indicate the places where \(y = 0\) and correlate with our solutions: \(x = 0\), \(x = \frac{1 + \sqrt{65}}{-8}\), and \(x = \frac{1 - \sqrt{65}}{-8}\).

Graphing confirms the accuracy of our analytical solution by demonstrating the real intersections of the polynomial with the x-axis.
A graph provides an intuitive understanding of behavior such as turning points and symmetry, facilitating a deeper comprehension beyond just numeric solutions.