Use the discriminant to determine the number of real solutions of $5 = 2x - x^2$.

1 answer

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.