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. Start with the given system of equations: \(y = x^{2} + 6x + 9\) and \(x + 2y = -2\).
2. Substitute the expression for \(y\) from the first equation into the second equation to eliminate \(y\). This gives: \(x + 2(x^{2} + 6x + 9) = -2\).
3. Expand and simplify the equation: \(x + 2x^{2} + 12x + 18 = -2\). Combine like terms to get a quadratic equation in terms of \(x\).
4. Bring all terms to one side to set the quadratic equation equal to zero: \$2x^{2} + 13x + 20 = 0$.
5. Solve the quadratic equation using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\), where \(a = 2\), \(b = 13\), and \(c = 20\). After finding the values of \(x\), substitute each back into the original equation for \(y\) to find the corresponding \(y\) values.

Key Concepts

Nonlinear Systems of Equations

A nonlinear system involves at least one equation that is not linear, such as quadratic or higher-degree polynomials. Solving these systems requires methods beyond simple substitution or elimination used for linear systems, often involving substitution of one equation into another to reduce the system to a single-variable equation.

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, including factoring or the quadratic formula.

Complex Solutions

When solving polynomial equations, solutions may include complex numbers if the discriminant is negative. Complex solutions have a real part and an imaginary part, and it is important to include these when the problem specifies all solutions, not just real ones.