To determine the number of real solutions for the equation \( 5 = 2x - x^2 \), we first rewrite it in standard quadratic form. We'll rearrange the equation:
\[ -x^2 + 2x - 5 = 0. \]
For convenience, we can multiply through by \(-1\) to make the coefficient of \(x^2\) positive:
\[ x^2 - 2x + 5 = 0. \]
Next, we identify the coefficients of the quadratic equation \( ax^2 + bx + c = 0 \):
- \( a = 1 \),
- \( b = -2 \),
- \( c = 5 \).
To determine the number of real solutions, we calculate the discriminant, given by the formula:
\[ D = b^2 - 4ac. \]
Substituting in our values for \(a\), \(b\), and \(c\):
\[ D = (-2)^2 - 4 \cdot 1 \cdot 5. \] \[ D = 4 - 20. \] \[ D = -16. \]
The discriminant \(D\) is negative (\(-16 < 0\)). According to the properties of the discriminant, a negative discriminant indicates that there are no real solutions to the quadratic equation.
Thus, the equation \( 5 = 2x - x^2 \) has
\[ \boxed{0} \]
real solutions.
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