Solve each equation. 8(x-4)⁴-10(x-4)²=-3

Verified step by step guidance

1. Start by recognizing that the equation has expressions with powers of the same base, \( (x-4) \). To simplify, use a substitution: let \( y = (x-4)^2 \). This means \( (x-4)^4 = y^2 \).
2. Rewrite the original equation \( 8(x-4)^4 - 10(x-4)^2 = -3 \) using the substitution \( y \): it becomes \( 8y^2 - 10y = -3 \).
3. Bring all terms to one side to set the equation equal to zero: \( 8y^2 - 10y + 3 = 0 \). This is a quadratic equation in terms of \( y \).
4. Solve the quadratic equation \( 8y^2 - 10y + 3 = 0 \) using the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a=8 \), \( b=-10 \), and \( c=3 \).
5. After finding the values of \( y \), substitute back \( y = (x-4)^2 \) and solve each resulting equation \( (x-4)^2 = y \) by taking the square root of both sides, remembering to consider both positive and negative roots.

Key Concepts

Substitution Method for Polynomial Equations

This method involves replacing a complex expression with a single variable to simplify the equation. For example, letting y = (x - 4)^2 transforms the quartic equation into a quadratic form, making it easier to solve.

Solving Quadratic Equations

Once the equation is simplified into a quadratic form, techniques such as factoring, completing the square, or using the quadratic formula can be applied to find the values of the variable.

Back-Substitution and Checking Solutions

After solving for the substituted variable, replace it back with the original expression to find x. It's important to check all solutions in the original equation to ensure they are valid and do not produce extraneous results.