Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5. y = x^2 + 6x + 9 x + 2y = -2

Verified step by step guidance

1. <Step 1: Identify the system of equations.> The system consists of two equations: $y=x^2+6x+9$ and $x+2y=-2$.
2. <Step 2: Substitute the expression for $y$ from the first equation into the second equation.> Replace $y$ in $x+2y=-2$ with $x^2+6x+9$.
3. <Step 3: Simplify the resulting equation.> This will give you a quadratic equation in terms of $x$.
4. <Step 4: Solve the quadratic equation.> Use the quadratic formula $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ to find the values of $x$.
5. <Step 5: Substitute the values of $x$ back into the first equation to find the corresponding $y$ values.> This will give you the solutions to the system, including any complex solutions.

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 this expression into the other equation. This method simplifies the system, allowing for easier solving of the equations. 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 include 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 at real points, indicating that the solutions may exist in the complex plane.