Solve the Parabola Inequality: y = -x²- 5x When f(x) < 0

Look at the following function:

$y=-x^2-5x$

Determine for which values of $x$ the following is true:

$f\left(x\right) < 0$

To determine where the function $y = -x^2 - 5x$ is negative, follow these steps:

• Step 1: Find the roots of the equation $-x^2 - 5x = 0$.
• Step 2: Factor the equation: $-x(x + 5) = 0$.
• Step 3: Solve for $x$:
  • $x = 0$
  • $x = -5$
• Step 4: Analyze the sign of the quadratic function in the intervals determined by the roots:
  • Interval 1: $x < -5$
  • Interval 2: $-5 < x < 0$
  • Interval 3: $x > 0$

Consider an example point in each interval to determine if $y$ is negative:

• For $x < -5$, choose $x = -6$:
  $y = -(-6)^2 - 5(-6) = -36 + 30 = -6$, which is negative.
• For $-5 < x < 0$, choose $x = -1$:
  $y = -(-1)^2 - 5(-1) = -1 + 5 = 4$, which is positive.
• For $x > 0$, choose $x = 1$:
  $y = -(1)^2 - 5(1) = -1 - 5 = -6$, which is negative.

Therefore, the intervals where $f(x) < 0$ are $x < -5$ and $x > 0$.

The correct answer is $x > 0$ or $x < -5$.