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

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Start with the given system of equations: $y = 6x + x^2$ and $4x - y = -3$.

Substitute the expression for $y$ from the first equation into the second equation to eliminate $y$. This gives: $4x - (6x + x^2) = -3$.

Simplify the equation obtained after substitution: $4x - 6x - x^2 = -3$ which simplifies to $-2x - x^2 = -3$.

Rewrite the equation in standard quadratic form by moving all terms to one side: $-x^2 - 2x + 3 = 0$ or equivalently $x^2 + 2x - 3 = 0$ (multiply both sides by -1).

Solve the quadratic equation $x^2 + 2x - 3 = 0$ using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a=1$, $b=2$, and $c=-3$. After finding the values of $x$, substitute each back into the original equation $y = 6x + x^2$ to find the corresponding $y$ values.

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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 or factoring to find all possible solutions.

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 solving quadratic equations.

Complex Solutions

When solving nonlinear equations, solutions may include nonreal complex numbers, especially if the quadratic equation has a negative discriminant. Understanding how to find and interpret complex solutions is essential for providing all possible answers to the system.

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5.y=6x+x2y=6x+x^24x−y=−34x-y=-3