Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5. y = 6x + x^2 4x - y = -3

Key Concepts

Nonlinear Equations

Nonlinear equations are equations in which the variable(s) are raised to a power greater than one or involve products of variables. Unlike linear equations, which graph as straight lines, nonlinear equations can produce curves, parabolas, or other complex shapes. Understanding how to manipulate and solve these equations is crucial for finding their intersections or solutions.

Substitution Method

The substitution method is a technique used to solve systems of equations by expressing one variable in terms of another and substituting it into the other equation. This method simplifies the system, making it easier to solve for the remaining variable. It is particularly useful in nonlinear systems where one equation can be easily rearranged.

Complex Solutions

Complex solutions arise when the solutions to an equation involve imaginary numbers, typically represented as 'a + bi', where 'a' is the real part and 'b' is the imaginary part. In the context of nonlinear systems, recognizing and finding these solutions is essential, especially when the equations do not intersect in the real number plane, indicating that solutions may exist in the complex plane.