Solve each equation in Exercises 41–60 by making an appropriate substitution. x - 13√x + 40 = 0

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Identify the substitution to simplify the equation. Since the equation contains both \( x \) and \( \sqrt{x} \), let \( t = \sqrt{x} \). This means \( x = t^2 \).

Rewrite the original equation \( x - 13\sqrt{x} + 40 = 0 \) in terms of \( t \). Substitute \( x = t^2 \) and \( \sqrt{x} = t \) to get \( t^2 - 13t + 40 = 0 \).

Recognize that the equation \( t^2 - 13t + 40 = 0 \) is a quadratic equation in standard form. Use factoring, completing the square, or the quadratic formula to solve for \( t \).

After finding the values of \( t \), recall that \( t = \sqrt{x} \). To find \( x \), square each solution for \( t \) to get \( x = t^2 \).

Check each solution for \( x \) in the original equation to ensure it is valid, especially since \( \sqrt{x} \) implies \( x \geq 0 \). Discard any extraneous solutions.

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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 simpler variable to transform the equation into a more manageable form. In this problem, letting √x = t simplifies the equation, making it easier to solve for t before back-substituting to find x.

Quadratic Equations

A quadratic equation is a second-degree polynomial equation typically in the form ax² + bx + c = 0. After substitution, the given equation becomes quadratic in terms of the new variable, allowing the use of factoring, completing the square, or the quadratic formula to find solutions.

Domain Restrictions for Square Roots

Since √x represents the principal (non-negative) square root, the variable x must be non-negative (x ≥ 0). Additionally, solutions obtained after substitution must be checked to ensure they satisfy this domain restriction to avoid extraneous roots.