To find the solutions to the given system of equations:
[tex]\[
\begin{cases}
y = 2x^2 - 5x - 7 \\
y = 2x + 2
\end{cases}
\][/tex]
we start by setting the two equations equal to each other:
[tex]\[
2x^2 - 5x - 7 = 2x + 2
\][/tex]
Now, we need to move all terms to one side to form a standard quadratic equation:
[tex]\[
2x^2 - 5x - 7 - 2x - 2 = 0
\][/tex]
Simplify this equation:
[tex]\[
2x^2 - 7x - 9 = 0
\][/tex]
Now we need to solve this quadratic equation. We can use the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 2 \)[/tex], [tex]\( b = -7 \)[/tex], and [tex]\( c = -9 \)[/tex].
Calculate the discriminant ([tex]\( b^2 - 4ac \)[/tex]):
[tex]\[
b^2 - 4ac = (-7)^2 - 4(2)(-9) = 49 + 72 = 121
\][/tex]
Since the discriminant is positive, there will be two real solutions. Now, apply the quadratic formula:
[tex]\[
x = \frac{-(-7) \pm \sqrt{121}}{2(2)} = \frac{7 \pm 11}{4}
\][/tex]
This gives us two solutions for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{7 + 11}{4} = \frac{18}{4} = 4.5
\][/tex]
and
[tex]\[
x = \frac{7 - 11}{4} = \frac{-4}{4} = -1
\][/tex]
Now that we have the [tex]\( x \)[/tex]-values, we need to find the corresponding [tex]\( y \)[/tex]-values using the linear equation [tex]\( y = 2x + 2 \)[/tex]:
For [tex]\( x = 4.5 \)[/tex]:
[tex]\[
y = 2(4.5) + 2 = 9 + 2 = 11
\][/tex]
For [tex]\( x = -1 \)[/tex]:
[tex]\[
y = 2(-1) + 2 = -2 + 2 = 0
\][/tex]
Therefore, the solutions to the system of equations are:
[tex]\[
(4.5, 11) \quad \text{and} \quad (-1, 0)
\][/tex]
Looking at the given options, the correct solution is:
[tex]\[
(4.5, 11) \text{ and } (-1, 0)
\][/tex]
So the answer is:
[tex]\[
(\textbf{4.5, 11) and (-1, 0)}
\][/tex]
