Solve each equation in Exercises 41–60 by making an appropriate substitution. $9x^4=25x^2-16$

Verified step by step guidance

1. Identify the substitution to simplify the equation. Notice that the equation involves terms with \(x^4\) and \(x^2\). Let \(u = x^2\), so that \(x^4 = (x^2)^2 = u^2\).
2. Rewrite the original equation \(9x^4 = 25x^2 - 16\) in terms of \(u\): it becomes \(9u^2 = 25u - 16\).
3. Bring all terms to one side to set the equation equal to zero: \(9u^2 - 25u + 16 = 0\).
4. Solve the quadratic equation \(9u^2 - 25u + 16 = 0\) using the quadratic formula \(u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a=9\), \(b=-25\), and \(c=16\).
5. After finding the values of \(u\), substitute back \(u = x^2\) and solve each resulting equation \(x^2 = u\) for \(x\) by taking the square root, remembering to consider both positive and negative roots.

Key Concepts

Polynomial Equations

Polynomial equations involve expressions with variables raised to whole-number exponents. Understanding how to manipulate and solve these equations, especially higher-degree polynomials like quartic (degree 4), is essential for finding their roots or solutions.

Substitution Method

The substitution method simplifies complex equations by replacing a part of the equation with a new variable. For example, substituting y = x² transforms a quartic equation into a quadratic one, making it easier to solve.

Solving Quadratic Equations

Once substitution reduces the equation to quadratic form, solving it involves techniques like factoring, completing the square, or using the quadratic formula. These methods help find the values of the substituted variable, which can then be back-substituted to find the original variable.