Solve the Quadratic Inequality y = -4x² + 24x: Which x-values Apply?

Look at the following function:

$y=-4x^2+24x$

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

$f(x) < 0$

To solve this problem, we'll follow these steps:

• Step 1: Find the roots by setting the function equal to zero: $-4x^2 + 24x = 0$.
• Step 2: Factor the equation. We get $-4x(x - 6) = 0$.
• Step 3: Solve for $x$ to find the roots: $x = 0$ and $x = 6$.
• Step 4: Analyze intervals defined by these roots: these intervals are $(-\infty, 0)$, $(0, 6)$, and $(6, \infty)$.
• Step 5: Test the sign of $y = -4x^2 + 24x$ in each interval:
• In $(-\infty, 0)$, choose $x = -1$: $-4(-1)^2 + 24(-1) = -4 - 24 = -28$, so $y < 0$.
• In $(0, 6)$, choose $x = 3$: $-4(3)^2 + 24(3) = -36 + 72 = 36$, so $y > 0$.
• In $(6, \infty)$, choose $x = 7$: $-4(7)^2 + 24(7) = -196 + 168 = -28$, so $y < 0$.
• Step 6: Compile the solution set based on where $y < 0$: the intervals are $(-\infty, 0)$ and $(6, \infty)$.

Therefore, the solution is that $f(x) < 0$ for $x > 6$ or $x < 0$.