Domain Analysis: Find Valid Inputs for y = 5x² - 9/16

To solve the problem of finding the positive and negative domains of the function $y = 5x^2 - \frac{9}{16}$, follow these steps:

• Step 1: Set the function equal to zero to find the critical points: $5x^2 - \frac{9}{16} = 0$.
• Step 2: Solve the equation for $x$:
  $5x^2 = \frac{9}{16}$
  Divide both sides by 5:
  $x^2 = \frac{9}{80}$
  Take the square root of both sides:
  $x = \pm \sqrt{\frac{9}{80}}$.
• Step 3: Simplify the expression further:
  $x = \pm \frac{3}{\sqrt{80}} = \pm \frac{3}{4\sqrt{5}} = \pm \frac{3\sqrt{5}}{20}$.
• Step 4: Identify intervals based on the roots where the function could be positive or negative.

The roots are $x = \frac{3\sqrt{5}}{20}$ and $x = -\frac{3\sqrt{5}}{20}$.
The quadratic opens upwards since the coefficient of $x^2$ is positive. Therefore, $y$ will be negative between the roots, i.e.,
For negative domain: $-\frac{3\sqrt{5}}{20} < x < \frac{3\sqrt{5}}{20}$.
For positive domain: $x < -\frac{3\sqrt{5}}{20}$ or $x > \frac{3\sqrt{5}}{20}$.

Verifying against the choices, the correct answer is:

$x < -\frac{3\sqrt{5}}{20} < x < \frac{3\sqrt{5}}{20}$

$x > \frac{3\sqrt{5}}{20}$ or $x > 0 : x < -\frac{3\sqrt{5}}{20}$

This matches choice 3 in the given options.