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

Ch. 1 - Equations and Inequalities

Lial13th EditionCollege AlgebraISBN: 9780136881063당신이 사용하는 게 아니라요?교과서 변경

모든 교과서

Lial 13th Edition

Ch. 1 - Equations and Inequalities

문제 98

2장, 문제 98

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

검증된 단계별 안내

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 \( (x-4)^4 = y^2 \).

Rewrite the original equation \( 8(x-4)^4 - 10(x-4)^2 = -3 \) using the substitution \( y \): it becomes \( 8y^2 - 10y = -3 \).

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 \( y \).

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 \( c=3 \).

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.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.