Solve the Quadratic Inequality for y = 2x² - 50: When is f(x) > 0?

To solve this problem, we need to determine where the function $y = 2x^2 - 50$ is greater than zero.

Step 1: Find the roots of the equation $2x^2 - 50 = 0$.

Step 1.1: Solve the equation:

• $2x^2 - 50 = 0$
• $2x^2 = 50$
• $x^2 = 25$
• $x = \pm 5$

The roots are $x = 5$ and $x = -5$. These are the points where the parabola touches the x-axis.

Step 2: Analyze intervals defined by the roots. The $x$-values divide the number line into three intervals: $x < -5$, $-5 < x < 5$, and $x > 5$.

Step 3: Test each interval to find where $2x^2 - 50 > 0$.

• For $x < -5$: Choose a test point, e.g., $x = -6$. Then $y = 2(-6)^2 - 50 = 72 - 50 = 22$. Since 22 is positive, $y > 0$ for $x < -5$.
• For $-5 < x < 5$: Choose a test point, e.g., $x = 0$. Then $y = 2(0)^2 - 50 = -50$. Since -50 is negative, $y < 0$ for $-5 < x < 5$.
• For $x > 5$: Choose a test point, e.g., $x = 6$. Then $y = 2(6)^2 - 50 = 72 - 50 = 22$. Since 22 is positive, $y > 0$ for $x > 5$.

Therefore, the intervals where $y = 2x^2 - 50 > 0$ are $x < -5$ and $x > 5$.

The solution is thus: $x > 5$ or $x < -5$, corresponding to choice 4.