Solve the Quadratic Inequality: When Is 2x² - 24x Greater than Zero?

To solve for the values of $x$ where $y = 2x^2 - 24x > 0$, we begin with the quadratic equation:

$y = 2x^2 - 24x$

First, factor the quadratic expression:

$y = 2x(x - 12)$

To find where this expression is greater than zero, first determine the zeros of the function by setting the equation to zero:

$2x(x - 12) = 0$

Solving for $x$, we find:

• $x = 0$
• $x = 12$

These zeros divide the number line into three intervals to test: $x < 0$, $0 < x < 12$, and $x > 12$.

Choose test points from each interval, such as $x = -1$, $x = 1$, and $x = 13$, to evaluate the sign of the expression $2x(x - 12)$:

• For $x = -1$: $2(-1)((-1) - 12) = 26$, thus positive.
• For $x = 1$: $2(1)((1) - 12) = -22$, thus negative.
• For $x = 13$: $2(13)((13) - 12) = 26$, thus positive.

From the above test results, $y = 2x(x - 12) > 0$ when $x < 0$ or $x > 12$.

Thus, the values of $x$ that satisfy $f(x) > 0$ are:

$x > 12$ or $x < 0$