Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components.
3x² - y² = 11
xy = 12

검증된 단계별 안내

Start with the given system of equations: \[3x^2 - y^2 = 11\] \[xy = 12\]

From the second equation, express one variable in terms of the other. For example, solve for \(y\): \[y = \frac{12}{x}\] (assuming \(x \neq 0\)).

Substitute the expression for \(y\) into the first equation to eliminate \(y\): \[3x^2 - \left(\frac{12}{x}\right)^2 = 11\]

Simplify the equation obtained after substitution: \[3x^2 - \frac{144}{x^2} = 11\] Multiply both sides by \(x^2\) to clear the denominator: \[3x^4 - 144 = 11x^2\]

Rewrite the equation as a quadratic in terms of \(x^2\): \[3x^4 - 11x^2 - 144 = 0\] Let \(u = x^2\), then solve the quadratic equation \[3u^2 - 11u - 144 = 0\] for \(u\). After finding \(u\), back-substitute to find \(x\), and then use \(y = \frac{12}{x}\) to find \(y\).

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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

12m

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주요 개념

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

Nonlinear Systems of Equations

A nonlinear system involves equations where variables are raised to powers other than one or multiplied together. Solving such systems requires methods beyond simple substitution or elimination used for linear systems, often involving factoring, substitution, or using algebraic identities.

Substitution Method

The substitution method involves solving one equation for one variable and substituting that expression into the other equation. This reduces the system to a single equation with one variable, which can then be solved using algebraic techniques.

Complex Number Solutions

When solving nonlinear systems, solutions may include complex numbers, especially if the equations lead to negative values under square roots. Complex solutions have a real part and an imaginary part, and recognizing them is essential for providing all possible solutions.

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. 3x2 - y2 = 11 xy = 12