Solve each equation. 4(x+1)⁴-13(x+1)²=-9

Verified step by step guidance

1. Start by making a substitution to simplify the equation. Let \(y = (x+1)^2\). This transforms the original equation \(4(x+1)^4 - 13(x+1)^2 = -9\) into \(4y^2 - 13y = -9\).
2. Rewrite the equation in standard quadratic form by moving all terms to one side: \(4y^2 - 13y + 9 = 0\).
3. Solve the quadratic equation \(4y^2 - 13y + 9 = 0\) using the quadratic formula: \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a=4\), \(b=-13\), and \(c=9\).
4. After finding the values of \(y\), recall that \(y = (x+1)^2\). For each solution \(y_i\), solve the equation \((x+1)^2 = y_i\) by taking the square root of both sides: \(x+1 = \pm \sqrt{y_i}\).
5. Finally, solve for \(x\) by isolating it: \(x = -1 \pm \sqrt{y_i}\). These values of \(x\) are the solutions to the original equation.

Key Concepts

Substitution Method

The substitution method involves replacing a complex expression with a single variable to simplify the equation. In this problem, letting y = (x + 1)^2 transforms the quartic equation into a quadratic form, making it easier to solve.

Solving Quadratic Equations

Once the equation is rewritten as a quadratic in terms of y, techniques such as factoring, completing the square, or using the quadratic formula can be applied to find the values of y. These solutions are then used to find x.

Back-Substitution and Solving for x

After finding the values of y, substitute back y = (x + 1)^2 to solve for x. This typically involves taking square roots and considering both positive and negative roots to find all possible solutions.