Question

Which of the following best characterizes the number of real solutions for the system of equations y=−2x+1 and y=−x2−1?(1 point)
Responses

The system of equations has one real solution.
The system of equations has one real solution.

The system of equations has an infinite number of real solutions.
The system of equations has an infinite number of real solutions.

The system of equations has no real solutions
The system of equations has no real solutions

The system of equations has two real solutions.

Answer 1

To analyze the number of real solutions for the system of equations \( y = -2x + 1 \) and \( y = -x^2 - 1 \), we can set the two equations equal to each other since they both equal \( y \):

\[
-2x + 1 = -x^2 - 1
\]

Now, rearranging the equation gives:

\[
-x^2 + 2x + 1 + 1 = 0
\]

This simplifies to:

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

Multiplying through by -1 to make the quadratic standard gives:

\[
x^2 - 2x - 2 = 0
\]

Next, we can use the discriminant to determine the number of real solutions. For a quadratic equation of the form \( ax^2 + bx + c = 0 \), the discriminant \( D \) is given by:

\[
D = b^2 - 4ac
\]

For our equation \( x^2 - 2x - 2 = 0 \):

- \( a = 1 \)
- \( b = -2 \)
- \( c = -2 \)

Calculating the discriminant:

\[
D = (-2)^2 - 4(1)(-2) = 4 + 8 = 12
\]

Since the discriminant \( D = 12 \) is positive, this indicates that there are two distinct real solutions for the equation.

Thus, the best characterization of the number of real solutions for the system of equations is:

**The system of equations has two real solutions.**