Question

14
Multiple Choice 1 point
Solve the system of equations.
y=-4x
y=x2-12
• (-2,8) and (6,-24)
(-6, -24) and (2,8)
(-6, 24) and (2, -8)
(-2, -8) and (6, 24)

Answer 1

To solve the system of equations given by \( y = -4x \) and \( y = x^2 - 12 \), we can set these two equations equal to each other:

\[
-4x = x^2 - 12
\]

Rearranging this equation gives:

\[
x^2 + 4x - 12 = 0
\]

Now we can use the quadratic formula to find the values of \( x \):

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

where \( a = 1, b = 4, c = -12 \).

Calculating the discriminant:

\[
b^2 - 4ac = 4^2 - 4(1)(-12) = 16 + 48 = 64
\]

Now plug this back into the quadratic formula:

\[
x = \frac{-4 \pm \sqrt{64}}{2 \cdot 1} = \frac{-4 \pm 8}{2}
\]

This gives us two solutions:

1. \( x = \frac{4}{2} = 2 \)
2. \( x = \frac{-12}{2} = -6 \)

Now, we will find the corresponding \( y \) values for both \( x \) values using the equation \( y = -4x \):

1. When \( x = 2 \):
\[
y = -4(2) = -8 \quad \Rightarrow \quad (2, -8)
\]

2. When \( x = -6 \):
\[
y = -4(-6) = 24 \quad \Rightarrow \quad (-6, 24)
\]

Thus, the solutions to the system of equations are \( (-6, 24) \) and \( (2, -8) \).

Therefore, the correct answer is:

**(-6, 24) and (2, -8)**.