Solve the Quadratic Inequality: -x² + 4x > 0

To understand how to solve the inequality $-x^2 + 4x > 0$, let's break it down step-by-step:

• Step 1: Solve the related quadratic equation. Begin with $-x^2 + 4x = 0$. We can factor this as $x(-x + 4) = 0$, giving roots $x = 0$ and $x = 4$.
• Step 2: Use these roots to divide the real number line into intervals: $(-∞, 0)$, $(0, 4)$, and $(4, ∞)$.
• Step 3: Test a point from each interval in the inequality:
• For $(-∞, 0)$, choose $x = -1$: $-(-1)^2 + 4(-1) = -1 - 4 = -5$ (negative)
• For $(0, 4)$, choose $x = 2$: $-(2)^2 + 4(2) = -4 + 8 = 4$ (positive)
• For $(4, ∞)$, choose $x = 5$: $-(5)^2 + 4(5) = -25 + 20 = -5$ (negative)
• Step 4: Identify where $-x^2 + 4x > 0$: Only in the interval $(0, 4)$.

Therefore, the solution to the inequality $-x^2 + 4x > 0$ is the interval $(0, 4)$. This corresponds to the choice: $0 < x < 4$.