In Exercises 91–100, find all values of x satisfying the given conditions. y=x−x−2andy=4y = x - \sqrt{x - 2} \quad \text{and} \quad y = 4 y=x−x−2andy=4

Ch. 1 - Equations and Inequalities

Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경

모든 교과서

Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

문제 94

2장, 문제 94

In Exercises 91–100, find all values of x satisfying the given conditions. $y = x - $\sqrt{x - 2}$ $\quad$ $\text{and}$ $\quad$ y = 4$

검증된 단계별 안내

Start with the given system of equations: $y = x - \sqrt{x - 2}$ and $y = 4$.

Since both expressions equal $y$, set them equal to each other: $4 = x - \sqrt{x - 2}$.

Isolate the square root term: $\sqrt{x - 2} = x - 4$.

Square both sides to eliminate the square root: $(\sqrt{x - 2})^2 = (x - 4)^2$, which simplifies to $x - 2 = (x - 4)^2$.

Expand the right side and rearrange the equation to form a quadratic: $x - 2 = x^2 - 8x + 16$, then bring all terms to one side to get $0 = x^2 - 9x + 18$.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

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

Solving Equations Involving Square Roots

When an equation includes a square root, isolate the root expression and then square both sides to eliminate the root. This process may introduce extraneous solutions, so all potential solutions must be checked in the original equation.

Domain Restrictions for Square Root Functions

The expression inside a square root must be non-negative for real-valued functions. For y = x - √(x - 2), the domain requires x - 2 ≥ 0, so x ≥ 2. This restriction limits the possible values of x when solving the equation.

Checking Solutions for Extraneous Roots

After solving equations involving square roots, some solutions may not satisfy the original equation due to the squaring step. Substitute each solution back into the original equation to verify its validity and discard any extraneous roots.

In Exercises 91–100, find all values of x satisfying the given conditions. y=x−x−2andy=4y = x - \(\sqrt{x - 2}\) \(\quad\) \(\text{and}\) \(\quad\) y = 4 y=x−x−2andy=4

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Solving Equations Involving Square Roots

Domain Restrictions for Square Root Functions

Checking Solutions for Extraneous Roots

In Exercises 91–100, find all values of x satisfying the given conditions. $y = x - \(\sqrt{x - 2}\) \(\quad\) \(\text{and}\) \(\quad\) y = 4$