Answer
The solution is $(-2, 4)$.
Work Step by Step
We will use substitution to solve this system of equations. We substitute one of the expressions for $y$, which would mean that we are going to set the two equations equal to one another to solve for $x$ first:
$x^2 + 3x + 6 = -x + 2$
We want to move all terms to the left side of the equation.
$x^2 + 3x + x + 6 - 2 = 0$
Combine like terms:
$x^2 + 4x + 4 = 0$
Factor the quadratic equation. The quadratic equation takes the form $ax^2 + bx + c = 0$. We need to find factors whose product is $ac$ but sum up to $b$. In this exercise, $ac = 4$ and $b = 4$. The factors $2$ and $2$ will work:
$(x + 2)(x + 2) = 0$
The two factors are the same. Set the factor equal to $0$.
$x + 2 = 0$
Subtract $2$ from each side of the equation:
$x = -2$
Now that we have the value for $x$, we can plug it into one of the original equations to find the corresponding $y$ value. Let's use the second equation:
$y = -x + 2$
Substitute the solution $-2$ for $x$:
$y = -(-2) + 2$
Multiply next:
$y = 2 + 2$
Add to solve:
$y = 4$
The solution is $(-2, 4)$.
