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 \( 5 = 2x - x^2 \).
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