Solve the Quadratic Inequality: When Does y = -4x^2 + 24x Become Positive?

Look at the following function:

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

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

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

To determine the values of $x$ for which the function $y = -4x^2 + 24x$ is greater than zero, we will proceed as follows:

Step 1: Find the roots of the quadratic equation.

We start by solving $-4x^2 + 24x = 0$ to find the critical points. This can be factored as:

$x(-4x + 24) = 0$

This equation gives us two roots:

1. $x = 0$
2. $-4x + 24 = 0 \Rightarrow x = 6$

Step 2: Determine intervals for positivity.

The roots divide the number line into three intervals: $(-\infty, 0)$, $(0, 6)$, and $(6, \infty)$.

Since the parabola opens downwards (as indicated by the negative leading coefficient), the function will be positive between the roots:

$0 < x < 6$

Conclusion: To ensure the function $y = -4x^2 + 24x$ is greater than zero, the value of $x$ must be between 0 and 6.

Therefore, the solution to the problem is $0 < x < 6$.