Problem 22 - 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. $$\begin{aligned} &9 x^{4}+35 x^{2}-4=0\\\ &[-3,3] \text { by }[-10,100] \end{aligned}$$

Step-by-Step Solution

Answer

Solutions are \(x = \pm\sqrt{\frac{-35 + \sqrt{1441}}{18}}\) and \(x = \pm\sqrt{\frac{-35 - \sqrt{1441}}{18}}\).

Step 1: Recognize the Equation Type

The given equation \(9x^4 + 35x^2 - 4 = 0\) is a polynomial in \(x\). Notice that \(9x^4\) can be rewritten as \((3x^2)^2\), suggesting this is a quadratic in form of \(u = x^2\). This suggests using substitution.

Step 2: Substitute to Simplify

Substitute \(u = x^2\) so the equation becomes \(9u^2 + 35u - 4 = 0\). This transforms our quartic equation in \(x\) into a quadratic in \(u\).

Step 3: Solve the Quadratic Equation

Use the quadratic formula \(u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) with \(a = 9\), \(b = 35\), and \(c = -4\). Calculate the discriminant \(b^2 - 4ac = 35^2 - 4 \times 9 \times (-4) = 1441\). Thus, \(u = \frac{-35 \pm \sqrt{1441}}{18}\).

Step 4: Find Values of \(u\)

Calculate each value of \(u\): \(u_1 = \frac{-35 + \sqrt{1441}}{18}\) and \(u_2 = \frac{-35 - \sqrt{1441}}{18}\). These values of \(u\) should be simplified to get their exact forms.

Step 5: Solve for \(x\) with each \(u\) Value

Recall \(u = x^2\) and solve for \(x\). This means for each \(u\) value, we have \(x^2 = u\), leading to \(x = \pm\sqrt{u}\). Substitute \(u_1\) and \(u_2\) to get \(x = \pm\sqrt{\frac{-35 + \sqrt{1441}}{18}}\) and \(x = \pm\sqrt{\frac{-35 - \sqrt{1441}}{18}}\).

Step 6: Verify and Graph Real Solutions

Graph \(y_1 = 9x^4 + 35x^2 - 4\) using the window \([-3,3]\) for \(x\) and \([-10,100]\) for \(y\). Only consider real parts from the solutions found. Use a calculator to confirm the intersection with the x-axis (real solutions).

Key Concepts

Polynomial Equations

A polynomial equation is any equation that involves an expression made up of variables raised to various powers, added or subtracted together, and sometimes multiplied by constant coefficients. For example, the polynomial equation provided in the exercise is a quartic, or fourth-degree polynomial, written as:

• \(9x^4 + 35x^2 - 4 = 0\)

This type of equation includes four terms if expanded fully and looks similar to a quadratic equation in form, which can make it more daunting at first glance. However, recognizing that it can be rewritten using substitutions often simplifies the problem. For instance, by letting \(u = x^2\), we effectively reduce the quartic to:

• \(9u^2 + 35u - 4 = 0\)

This substitution is an essential technique when dealing with higher-degree polynomial equations, making them easier to handle because it turns the equation into a quadratic form that's usually more familiar from earlier math courses.

Quadratic Forms

A quadratic form is a specific type of polynomial where the highest power of the variable is two. This form is typically much easier to solve than higher-degree polynomials. In our exercise, once we applied the substitution \(u = x^2\), we changed our quartic equation into a quadratic equation:

• \(9u^2 + 35u - 4 = 0\)

To solve a quadratic equation, the quadratic formula

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

is often used, where \(a\), \(b\), and \(c\) are coefficients of the quadratic equation. Applying it here allows us to find the values of \(u\), which can then be back-substituted to find \(x\). This process underscores how quadratic forms can be crucial stepping stones in solving complex equations. The discriminant, \(b^2 - 4ac\), tells us the nature of the solutions—whether they are real or complex, helping us further understand what to expect.

Graphing Calculators

A graphing calculator is a powerful tool that can assist in solving equations by visualizing them. It allows for plotting functions to check for real solutions and provides insight into the behavior of polynomial equations. In this particular exercise, you would graph the left side of the equation:

• \(y_{1} = 9x^4 + 35x^2 - 4\)

Using the specified window of

• \([-3, 3]\) for the x-axis,
• \([-10, 100]\) for the y-axis

helps in identifying the real roots, as these intersections with the x-axis will indicate potential real solutions. Verifying solutions with a graphing calculator provides a visual confirmation of your analytic work, reinforcing both the accuracy and understanding of the solutions derived algebraically. They are particularly useful because they offer immediate feedback and can handle complex calculations that might be time-consuming by hand.