Problem 37 - College Algebra with Corequisite Support

Question

For the following exercises, solve the equation by identifying the quadratic form. Use a substitute variable and find all real solutions by factoring. $$ x^{4}-10 x^{2}+9=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = 1, -1, 3, -3\).

Step 1: Identify the Quadratic Form

The given equation is \(x^{4} - 10x^{2} + 9 = 0\). Notice that this resembles a quadratic equation in the form \(ax^2 + bx + c = 0\), but with \(x^4\) as the highest degree term. You can substitute \(u = x^2\), which leads to \(u^2 = x^4\). This substitution will transform the equation into a quadratic equation: \(u^2 - 10u + 9 = 0\).

Step 2: Factor the Quadratic Equation

With the substitution, we now have \(u^2 - 10u + 9 = 0\). To factor this, we look for two numbers that multiply to 9 and add to -10. The numbers are -1 and -9. Therefore, we can factor the equation as \((u - 1)(u - 9) = 0\).

Step 3: Solve for the Substitute Variable

Set each factor equal to zero: \(u - 1 = 0\) and \(u - 9 = 0\). Solving these gives \(u = 1\) and \(u = 9\).

Step 4: Back-Substitute to Solve for x

Recall that \(u = x^2\). Substitute back for each value of \(u\): - If \(u = 1\), then \(x^2 = 1\). Solving for \(x\), we get \(x = \pm 1\). - If \(u = 9\), then \(x^2 = 9\). Solving for \(x\), we get \(x = \pm 3\).

Step 5: List All Solutions

Combining all solutions, the real solutions to the equation are \(x = 1, -1, 3, -3\).

Key Concepts

Factoring Quadratic Equations

Factoring is a process used in mathematics to break down expressions into simpler parts called factors. When we factor a quadratic equation, we aim to express it as the product of two binomials. This is particularly useful for finding its roots or solutions. In the exercise above, the original equation can be rearranged into a quadratic form, namely, \( u^2 - 10u + 9 = 0 \). Here, the goal is to identify two numbers that multiply to the constant term, 9, and add to the coefficient of the linear term, -10.

To do this:

• We recognize that the numbers -1 and -9 multiply to give 9.
• Additionally, -1 and -9 add up to -10, satisfying the requirement.

Thus, the equation can be expressed in factored form as \((u - 1)(u - 9) = 0\). This factoring step simplifies the process of solving for the roots of the equation by making it possible to apply the Zero Product Property, which states that if a product equals zero, then at least one of the factors must be zero.

Using the Substitution Method

The substitution method is a helpful algebraic tool that simplifies complex equations by introducing a substitute variable. By changing variables, you can convert a challenging problem into a more familiar form, like turning a quartic equation into a quadratic one. In this problem, recognizing that the given equation \(x^4 - 10x^2 + 9 = 0\) can resemble a simple quadratic expression is key.

Here's how it's applied:

• Set \(u = x^2 \), which implies \(u^2 = x^4\).
• Make this substitution in the original equation to get \( u^2 - 10u + 9 = 0 \).

With this substitution, the equation is now much simpler to solve using methods appropriate for quadratic equations, such as factoring. After finding the solutions for \(u\), the process includes back-substituting \(u = x^2\) to solve for original variable \(x\), ensuring that all potential solutions are captured.

Finding Real Solutions

When solving equations, particularly quadratic equations, seeking real solutions means finding the values of the variable that satisfy the equation and are real numbers (not involving imaginary or complex numbers). The exercise here illustrates how to determine real solutions effectively by employing factoring and substitution.

To conclude with real solutions:

• Factor the quadratic form and discover the values of the substitute variable \( u \) that make each factor zero, resulting in \( u = 1 \) and \( u = 9 \).
• Substitute back to find \( x^2 = 1 \) and \( x^2 = 9 \).
• Solve these equations to find \( x = \pm 1 \) and \( x = \pm 3 \).

Thus, the real solutions to the original equation are \( x = 1, -1, 3, \) and \(-3\). This strategic approach ensures that all possibilities are covered, and the solutions found are valid within the set of real numbers. Understanding these steps and approaching problems systematically helps build confidence in solving various types of algebraic equations.