Solving x²-1/4 > 0: Finding Positive Domains of a Quadratic Function

To find when the function $y = x^2 - \frac{1}{4}$ is positive, we set up the inequality:
$x^2 - \frac{1}{4} > 0$

First, solve the equation $x^2 - \frac{1}{4} = 0$:
$x^2 = \frac{1}{4}$

Taking the square root of both sides, we get:

• $x = \frac{1}{2}$ or $x = -\frac{1}{2}$

These roots, $x = \frac{1}{2}$ and $x = -\frac{1}{2}$, partition the real number line into three intervals: $(-\infty, -\frac{1}{2})$, $(-\frac{1}{2}, \frac{1}{2})$, and $(\frac{1}{2}, \infty)$.

Now, we test each interval:

• For $x \in (-\infty, -\frac{1}{2})$, choose $x = -1$:
  $x^2 - \frac{1}{4} = 1 - \frac{1}{4} = \frac{3}{4} > 0$
• For $x \in (-\frac{1}{2}, \frac{1}{2})$, choose $x = 0$:
  $x^2 - \frac{1}{4} = 0 - \frac{1}{4} = -\frac{1}{4} < 0$
• For $x \in (\frac{1}{2}, \infty)$, choose $x = 1$:
  $x^2 - \frac{1}{4} = 1 - \frac{1}{4} = \frac{3}{4} > 0$

Therefore, the function is positive in the intervals $(-\infty, -\frac{1}{2})$ and $(\frac{1}{2}, \infty)$.

Thus, the solution for the inequality $x^2 - \frac{1}{4} > 0$ is:
$x > \frac{1}{2}$ or $x < -\frac{1}{2}$

From the multiple-choice options, the correct answer is choice 3.