Determine X for Negative Values of the Function: y=2x²-24x

To solve the problem, follow these steps:

• Step 1: Solve the quadratic equation $2x^2 - 24x = 0$.
• Step 2: Factor the equation as $2x(x - 12) = 0$.
• Step 3: Determine the roots, which are $x = 0$ and $x = 12$.
• Step 4: Analyze the intervals determined by these roots: $x < 0$, $0 < x < 12$, and $x > 12$.
• Step 5: Test each interval to see where $f(x) < 0$.

Now let's work through each step:

Step 1: Solve the equation $2x(x - 12) = 0$. This gives us roots at $x = 0$ and $x = 12$.

Step 2: The quadratic can be negative between the roots, so we consider the interval $0 < x < 12$.

Step 3: Test the sign of $2x(x - 12)$ in each interval:

• Interval $(-\infty, 0)$: Choose a test point $x = -1$. The expression is $2(-1)((-1) - 12) = 2(-1)(-13) = 26$, which is positive.
• Interval $(0, 12)$: Choose a test point $x = 1$. The expression is $2(1)(1 - 12) = 2(1)(-11) = -22$, which is negative.
• Interval $(12, \infty)$: Choose a test point $x = 13$. The expression is $2(13)(13 - 12) = 2(13)(1) = 26$, which is positive.

Thus, the quadratic is negative in the interval $0 < x < 12$.

Therefore, the solution to the problem is $0 < x < 12$.