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

Verified step by step guidance

1. Start with the given system of equations: \[3x^2 - y^2 = 11\] \[xy = 12\]
2. From the second equation, express one variable in terms of the other. For example, solve for \(y\): \[y = \frac{12}{x}\] (assuming \(x \neq 0\)).
3. Substitute the expression for \(y\) into the first equation to eliminate \(y\): \[3x^2 - \left(\frac{12}{x}\right)^2 = 11\]
4. Simplify the equation obtained after substitution: \[3x^2 - \frac{144}{x^2} = 11\] Multiply both sides by \(x^2\) to clear the denominator: \[3x^4 - 144 = 11x^2\]
5. 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 \(y\).

Key Concepts

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.