Determine X Values for Positive Outputs in y=-x²-5x

To determine for which values of $x$ the quadratic function is positive, follow these steps:

• Step 1: Set the quadratic equation to zero: $-x^2 - 5x = 0$.
• Step 2: Factor the equation: $-x(x + 5) = 0$.
• Step 3: Solve for the roots: $x = 0$ and $x = -5$.
• Step 4: Determine the intervals defined by the roots: $(-\infty, -5)$, $(-5, 0)$, and $(0, \infty)$.
• Step 5: Test each interval:
• For interval $(-\infty, -5)$, choose $x = -6$: $-(-6)^2 - 5(-6) = -36 + 30 = -6$ (negative).
• For interval $(-5, 0)$, choose $x = -1$: $-(-1)^2 - 5(-1) = -1 + 5 = 4$ (positive).
• For interval $(0, \infty)$, choose $x = 1$: $-(1)^2 - 5(1) = -1 - 5 = -6$ (negative).
• Step 6: The function is positive in the interval $(-5, 0)$.

Thus, the solution to the inequality $f(x) > 0$ is $-5 < x < 0$.