Solve the Quadratic Inequality: -x² + 2x > 0

To solve the inequality $-x^2 + 2x > 0$, we begin by considering the corresponding equation $-x^2 + 2x = 0$.

First, factor the quadratic equation:

• Rearrange the terms: $-x^2 + 2x = 0$ becomes $x(2 - x) = 0$.
• This gives us the roots $x = 0$ and $x = 2$.

These roots divide the number line into three intervals: $x < 0$, $0 < x < 2$, and $x > 2$.

We need to test these intervals to determine where the inequality holds:

• For $x < 0$, choose a test point like $x = -1$: the expression $-(-1)^2 + 2(-1) = -1 - 2 = -3$, which is not greater than zero.
• For $0 < x < 2$, choose a test point like $x = 1$: the expression $-(1)^2 + 2(1) = -1 + 2 = 1$, which is greater than zero.
• For $x > 2$, choose a test point like $x = 3$: the expression $-(3)^2 + 2(3) = -9 + 6 = -3$, which is not greater than zero.

Thus, the inequality $-x^2 + 2x > 0$ is satisfied for the interval $0 < x < 2$.

Therefore, the solution to the inequality is $\mathbf{0 < x < 2}$, which corresponds to choice 2 in the given options.