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

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 95

2장, 문제 95

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

검증된 단계별 안내

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 \(4y^2 - 13y = -9\).

Rewrite the equation in standard quadratic form by moving all terms to one side: \(4y^2 - 13y + 9 = 0\).

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 \(c=9\).

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: \(x+1 = \pm \sqrt{y_i}\).

Finally, solve for \(x\) by isolating it: \(x = -1 \pm \sqrt{y_i}\). These values of \(x\) are the solutions to the original equation.

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

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.