Solve each equation in Exercises 41–60 by making an appropriate substitution. (x - 5)² - 4(x - 5) - 21 = 0

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Identify the substitution to simplify the equation. Let \( u = x - 5 \). This transforms the equation into a quadratic in terms of \( u \).

Rewrite the original equation using the substitution: \( (x - 5)^2 - 4(x - 5) - 21 = 0 \) becomes \( u^2 - 4u - 21 = 0 \).

Solve the quadratic equation \( u^2 - 4u - 21 = 0 \) by factoring, completing the square, or using the quadratic formula.

After finding the values of \( u \), substitute back \( u = x - 5 \) to get equations in terms of \( x \).

Solve each resulting linear equation for \( x \) to find the solutions to the original equation.

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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 - 5) transforms the equation into a quadratic in terms of y, making it easier to solve.

Quadratic Equations

A quadratic equation is a second-degree polynomial equation of the form ax² + bx + c = 0. Recognizing the transformed equation as quadratic allows the use of factoring, completing the square, or the quadratic formula to find solutions.

Back-Substitution

After solving the quadratic equation for the substituted variable, back-substitution involves replacing the variable with the original expression to find the values of x. This step ensures the solutions correspond to the original equation.

Solve each equation in Exercises 41–60 by making an appropriate substitution. (x - 5)2 - 4(x - 5) - 21 = 0