Answer
$(-1, 0)$ and $\left(\frac{1}{4}, \frac{25}{16}\right)$
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:
$-3x^2 - x + 2 = x^2 + 2x + 1$
We want to move all terms to the left side of the equation.
First, we will subtract $x^2$ from both sides of the equation:
$-4x^2 - x + 2 = 2x + 1$
Subtract $2x$ from both sides of the equation:
$-4x^2 - 3x + 2 = 1$
Subtract $1$ from both sides of the equation:
$-4x^2 - 3x + 1 = 0$
Let's factor this quadratic equation by splitting the middle term. We have a quadratic equation, which is in the form $ax^2 + bx + c = 0$. We need to find which factors multiplied together will equal $ac$ but when added together will equal $b$.
In this equation, $ac$ is $-4$ and $b$ is $-3$. Let's look at the possibilities:
$-4$ and $1$
$-2$ and $2$
The first option will work.
Let's rewrite the equation and split the middle term using these two factors:
$-4x^2 - 4x + x + 1 = 0$
Group the first two and last two terms:
$(-4x^2 - 4x) + (x + 1) = 0$
Factor common terms out:
$-4x(x + 1) + (x + 1) = 0$
Group the factors:
$(-4x + 1)(x + 1) = 0$
Set each factor equal to $0$.
First factor:
$-4x + 1 = 0$
Subtract $1$ from each side of the equation:
$-4x = -1$
Divide each side of the equation by $-4$:
$x = \frac{1}{4}$
Second factor:
$x + 1 = 0$
Subtract $1$ from each side of the equation:
$x = -1$
Now that we have the two possible values for $x$, we can plug them into one of the original equations to find the corresponding $y$ values. Let's use the second equation:
$y = (-1)^2 + 2(-1) + 1$
Evaluate the exponent first:
$y = 1 - 2 + 1$
Add or subtract from left to right:
$y = 0$
Let's solve for $y$ using the other solution for $x$:
$y = (\frac{1}{4})^2 + 2(\frac{1}{4}) + 1$
Evaluate the exponent first:
$y = \frac{1}{16} + 2(\frac{1}{4}) + 1$
Multiply next:
$y = \frac{1}{16} + \frac{2}{4} + 1$
Find equivalent fractions with common denominators to add the fractions together:
$y = \frac{1}{16} + \frac{8}{16} + \frac{16}{16}$
Add or subtract from left to right:
$y = \frac{25}{16}$
The solutions are $(-1, 0)$ and $\left(\frac{1}{4}, \frac{25}{16}\right)$.
