Problem 38 - Intermediate Algebra

Question

Solve each equation. $$ 4 x^{4}+11 x^{2}-45=0 $$

Step-by-Step Solution

Answer

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

Step 1: Substitute Variable

Notice that the equation is in a form where it can be simplified by using a substitution. Let \( y = x^{2} \). Then, the equation becomes \( 4y^{2} + 11y - 45 = 0 \).

Step 2: Solve the Quadratic Equation

Now, solve the quadratic equation \( 4y^{2} + 11y - 45 = 0 \). This can be done using the quadratic formula, where \( a = 4 \), \( b = 11 \), and \( c = -45 \). The quadratic formula is \( y = \frac{{-b \pm \sqrt{{b^{2} - 4ac}}}}{2a} \).

Step 3: Calculate Discriminant

First, calculate the discriminant: \( b^{2} - 4ac = 11^{2} - 4\times 4 \times (-45) = 121 + 720 = 841 \).

Step 4: Use Quadratic Formula

With the discriminant calculated, use the quadratic formula: \( y = \frac{{-11 \pm \sqrt{841}}}{8} \). This simplifies to \( y = \frac{{-11 + 29}}{8} \) or \( y = \frac{{-11 - 29}}{8} \).

Step 5: Solve for y

Calculate the two possible values for \( y \):1. \( y = \frac{{18}}{8} = \frac{9}{4} \).2. \( y = \frac{{-40}}{8} = -5 \).

Step 6: Re-substitute for x

Return to the original variable \( x \) using \( y = x^{2} \):1. If \( y = \frac{9}{4} \), then \( x^{2} = \frac{9}{4} \). Solving gives \( x = \pm \frac{3}{2} \).2. If \( y = -5 \), this doesn't provide a real solution for \( x \), as the square of a real number cannot be negative.

Step 7: Write Final Solutions

Therefore, the solutions to the original equation are \( x = \frac{3}{2} \) and \( x = -\frac{3}{2} \).

Key Concepts

Substitution Method

In the context of quadratic equations, the substitution method is a very handy technique. When an equation involves higher powers like a quartic term (e.g., \( x^4 \)), it might appear complex at first glance. However, by identifying patterns within the equation that resemble a simpler quadratic form, you can make a substitution to simplify it. For example, in the exercise \( 4x^4 + 11x^2 - 45 = 0 \), the substitution \( y = x^2 \) transforms the equation into a more familiar quadratic form: \( 4y^2 + 11y - 45 = 0 \). This substitution reduces the complexity of the equation and allows us to use techniques primarily associated with quadratics. By solving this new quadratic equation for \( y \), we can later substitute back to find the solutions in terms of the original variable \( x \). This method is particularly useful when the substitution results in a quadratic equation, enabling the application of well-known techniques like the quadratic formula.

Quadratic Formula

The quadratic formula is an essential tool when it comes to solving quadratic equations. It is designed to find the roots of any quadratic equation in the general form \( ax^2 + bx + c = 0 \). The formula is given by:

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

To use this formula, you need to identify the coefficients \( a \), \( b \), and \( c \) from the equation. In our example, substituting \( y = x^2 \) transformed the quartic equation to the quadratic \( 4y^2 + 11y - 45 = 0 \), where \( a = 4 \), \( b = 11 \), and \( c = -45 \). By plugging these values into the quadratic formula, it computes the possible values for \( y \), which ultimately leads to the solutions of the original problem. The use of the quadratic formula guarantees that you can solve for any quadratic, provided the discriminant is non-negative.

Discriminant

The discriminant is a critical concept when working with quadratic equations. It is the expression found under the square root in the quadratic formula: \( b^2 - 4ac \). The discriminant provides valuable information about the nature of the roots of the quadratic equation, such as:

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If it is zero, there is one real root or a repeated root.
• A negative discriminant indicates that the roots are complex and not real.

In the exercise, the discriminant of the quadratic equation \( 4y^2 + 11y - 45 = 0 \) was calculated as \( 11^2 - 4 \times 4 \times (-45) = 841 \). Since 841 is positive and a perfect square, it indicates that the quadratic equation has two distinct real roots. Understanding the discriminant helps predict the nature of the solutions without explicitly computing them, offering an efficient way to assess the solutions' characteristics before proceeding with further calculations.