Problem 19 - 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. $$\begin{aligned} &4 x^{4}-25 x^{2}+36=0\\\ &[-5,5] \text { by }[-5,100] \end{aligned}$$

Step-by-Step Solution

Answer

The complex solutions are \( x = \pm 2 \) and \( x = \pm \frac{3}{2} \).

Step 1: Substitute Variable

To simplify the equation, let us substitute a new variable. Substitute \( z = x^2 \) into the original equation. This transforms the quartic equation into a quadratic one: \( 4z^2 - 25z + 36 = 0 \).

Step 2: Use the Quadratic Formula

The equation \( 4z^2 - 25z + 36 = 0 \) can be solved using the quadratic formula: \( z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \( a = 4 \), \( b = -25 \), and \( c = 36 \). Substitute these into the formula to find \( z \).

Step 3: Calculate the Discriminant

Calculate the discriminant \( b^2 - 4ac \). This will be \( (-25)^2 - 4 \times 4 \times 36 = 625 - 576 = 49 \). Since the discriminant is positive, we have two real solutions for \( z \).

Step 4: Solve for \( z \)

Substitute the discriminant back into the quadratic formula: \[ z = \frac{25 \pm \sqrt{49}}{8} \]. Calculate the solutions, which gives \( z = \frac{25 + 7}{8} = 4 \) and \( z = \frac{25 - 7}{8} = \frac{9}{4} \).

Step 5: Find \( x \) Values

Recall that \( z = x^2 \). Thus we have two equations to solve: \( x^2 = 4 \) and \( x^2 = \frac{9}{4} \). Solve these to find \( x = \pm 2 \) and \( x = \pm \frac{3}{2} \).

Step 6: Verify the Real Solutions Graphically

Graph the original function \( y_1 = 4x^4 - 25x^2 + 36 \) in the viewing window \([-5,5]\) by \([-5,100]\). Verify that the x-intercepts occur at \( x = \pm 2 \) and \( x = \pm \frac{3}{2} \), supporting the calculated real solutions.

Key Concepts

Quartic Equation

A **quartic equation** is a polynomial equation of degree 4, which means it contains a term with the variable raised to the fourth power. In our problem, the quartic equation is given as \(4x^4 - 25x^2 + 36 = 0\). These types of equations can sometimes be tricky to solve directly, so simplifying them can make the process easier.

To solve a quartic equation, you might need to use techniques like substituting variables to transform it into a more straightforward quadratic equation. In this example, substituting \(z = x^2\) is a clever way to reduce the degree of the polynomial, changing the problem into something more manageable. This conversion simplifies the original fourth-degree polynomial into \(4z^2 - 25z + 36 = 0\), and from here, traditional techniques like the quadratic formula can be applied to solve for \(z\).

Quadratic Formula

The **quadratic formula** is a crucial tool used to find the roots of quadratic equations, which are polynomials in the form of \(ax^2 + bx + c = 0\). The formula is written as \(z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

In our transformed equation \(4z^2 - 25z + 36 = 0\), we identify \(a = 4\), \(b = -25\), and \(c = 36\). Plugging these values into the quadratic formula enables us to find potential solutions for \(z\). It's a powerful solution method because it covers all potential cases, including those with complex solutions. It provides an exact form for each solution, which is critical when you need precise calculations rather than approximate values. In our exercise, using the quadratic formula gave us \(z = 4\) and \(z = \frac{9}{4}\).

Discriminant

The **discriminant** is a part of the quadratic formula found under the square root sign, \(b^2 - 4ac\). It indicates the nature of the roots of a quadratic equation.

For the equation \(4z^2 - 25z + 36 = 0\), the discriminant is calculated to be \(49\). This value is significant because it tells us about the solutions' characteristics:

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

In our case, since the discriminant is positive, we can be confident about having two separate real solutions. Understanding the discriminant helps in anticipating what the solutions will look like, whether we'll end up having complex answers or sticking with real numbers.

Graphing Solutions

**Graphing solutions** is a practical way to validate your analytical results. In our problem, after solving the equation, graphing the polynomial \(y_1 = 4x^4 - 25x^2 + 36\) helps confirm the real roots. Using the specified viewing window ([-5,5] by [-5,100]) on a graphing calculator can provide a visual verification for the x-intercepts of the graph, which represent solutions to the equation.

The process of graphing allows you to see where the curve crosses the x-axis. Each intercept corresponds to a real solution. For this exercise, the graph should intersect the x-axis at \(x = \pm 2\) and \(x = \pm \frac{3}{2}\). This step not only supports the analytical solutions we've computed but also offers an intuitive understanding of how the polynomial behaves. It confirms that the computed solutions make sense within the problem's context.