Question

Use the discriminant to determine the number of real solutions of 5=2x−x2
.(1 point)

Answer 1

To use the discriminant to determine the number of real solutions to the equation \( 5 = 2x - x^2 \), we first need to rearrange the equation into the standard form of a quadratic equation \( ax^2 + bx + c = 0 \).

Starting with the given equation:

\[ 5 = 2x - x^2 \]

We can rearrange this to:

\[ x^2 - 2x + 5 = 0 \]

Now, we identify the coefficients for the quadratic equation:

- \( a = 1 \)
- \( b = -2 \)
- \( c = 5 \)

Next, we calculate the discriminant \( D \) using the formula:

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

Substituting the values of \( a \), \( b \), and \( c \):

\[ D = (-2)^2 - 4 \cdot 1 \cdot 5 \]
\[ D = 4 - 20 \]
\[ D = -16 \]

The discriminant \( D = -16 \) is less than zero.

According to the properties of the discriminant:

- If \( D > 0 \), there are two distinct real solutions.
- If \( D = 0 \), there is exactly one real solution (a repeated solution).
- If \( D < 0 \), there are no real solutions.

Since \( D = -16 < 0 \), we conclude that there are **no real solutions** to the equation \( 5 = 2x - x^2 \).