Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components.
x² - y = 0
x + y = 2

Verified step by step guidance

1. Start with the given system of equations: \[x^2 - y = 0\] \[x + y = 2\]
2. From the first equation, express \(y\) in terms of \(x\): \[y = x^2\]
3. Substitute this expression for \(y\) into the second equation: \[x + x^2 = 2\]
4. Rewrite the equation to standard quadratic form: \[x^2 + x - 2 = 0\]
5. Solve the quadratic equation using the quadratic formula or factoring to find the values of \(x\), then substitute back to find corresponding \(y\) values using \(y = x^2\).

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 substitution, factoring, or using the quadratic formula.

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, making it easier to solve, especially useful when one equation is already solved for a variable.

Complex Solutions

When solving nonlinear systems, solutions may include complex numbers if the equations lead to negative values under square roots or other operations. Understanding complex numbers and how to express solutions in terms of real and imaginary parts is essential for providing all possible solutions.