Question

No Real Solutions Quick Check
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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 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 one real solution.
The system of equations has one real solution.

The system of equations has two real solutions.

Answer 1

To determine 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 this equation gives:

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

Next, we can use the quadratic formula to find the solutions, where \( a = 1, b = -2, c = 2 \):

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
\[
x = \frac{2 \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot 2}}{2 \cdot 1}
\]
\[
x = \frac{2 \pm \sqrt{4 - 8}}{2}
\]
\[
x = \frac{2 \pm \sqrt{-4}}{2}
\]
\[
x = \frac{2 \pm 2i}{2}
\]
\[
x = 1 \pm i
\]

Since the discriminant \( b^2 - 4ac \) is negative (\( -4 \)), this indicates that there are no real solutions to the quadratic equation.

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

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