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

Answer 1

To determine the number of real solutions for the system of equations given by:

1. \( y = -2x + 1 \) (a linear equation)
2. \( y = -x^2 - 1 \) (a quadratic equation)

we can set the two equations equal to each other to find the points of intersection:

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

Rearranging this, we get:

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

Next, we will use the discriminant \(D\) from the quadratic formula \(ax^2 + bx + c = 0\):

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

For our equation:

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

Calculating the discriminant:

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

Since the discriminant \(D\) is negative, this indicates that there are no real solutions to the equation.

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

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