Solve the Quadratic Inequality: Determine x for which x² - 4 > 0

Look at the function below:

$y=x^2-4$

Then determine for which values of $x$ the following is true:

$f(x) > 0$

To find the solution to when $y = x^2 - 4$ is greater than zero, we start by analyzing the inequality:

$x^2 - 4 > 0$

This expression can be factored as:

$(x - 2)(x + 2) > 0$

The roots of the equation $(x - 2)(x + 2) = 0$ are $x = 2$ and $x = -2$. These points divide the real number line into intervals. We need to determine on which intervals the expression is positive. Let's analyze the intervals:

• $x < -2$: Choose $x = -3$, then both factors are negative: $(x - 2) < 0$ and $(x + 2) < 0$, making their product $(x - 2)(x + 2) > 0$.
• $-2 < x < 2$: Choose $x = 0$, then one factor is positive and the other is negative: $(x - 2) < 0$ and $(x + 2) > 0$, making their product negative.
• $x > 2$: Choose $x = 3$, then both factors are positive: $(x - 2) > 0$ and $(x + 2) > 0$, making their product positive.

Therefore, the function $x^2 - 4$ is positive for $x > 2$ or $x < -2$.

Thus, the solution to the inequality $y > 0$ is:

$x > 2$ or $x < -2$.