Question

use the discriminant to determine the number of real solutions of $5 = 2x - x^2$. (1 point)

$\square$ real solution(s)

Explanation:

Step1: Rewrite the equation in standard form

First, we rewrite the equation \(5 = 2x - x^2\) in the standard quadratic form \(ax^2+bx + c = 0\). By rearranging the terms, we get \(x^2-2x + 5=0\). Here, \(a = 1\), \(b=- 2\) and \(c = 5\).

Step2: Calculate the discriminant

The formula for the discriminant (\(D\)) of a quadratic equation \(ax^2+bx + c = 0\) is \(D=b^2-4ac\). Substituting the values of \(a\), \(b\) and \(c\) we found above:
\(D=(-2)^2-4\times1\times5\)
\(D = 4 - 20\)
\(D=- 16\)

Step3: Determine the number of real solutions

For a quadratic equation, if the discriminant \(D>0\), there are two distinct real solutions; if \(D = 0\), there is one real solution (a repeated root); if \(D<0\), there are no real solutions (two complex conjugate solutions). Since \(D=-16<0\), the equation has no real solutions.

Answer:

\(0\)