Uncover Function Domains: Analyzing y=-2x²-8x-8

To find the positive and negative domains of the quadratic function $y = -2x^2 - 8x - 8$, we first find the roots of the equation by using the quadratic formula.

The quadratic formula is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

From the equation $-2x^2 - 8x - 8 = 0$, we identify:

• $a = -2$
• $b = -8$
• $c = -8$

Now substitute these into the quadratic formula:

$x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(-2)(-8)}}{2(-2)}$

This simplifies to:

$x = \frac{8 \pm \sqrt{64 - 64}}{-4}$

Which further simplifies to:

$x = \frac{8 \pm \sqrt{0}}{-4}$

Thus,

$x = \frac{8}{-4}$

The roots are:

$x = -2$

Since this is a parabola opening downwards (because $a < 0$), it is positive between its roots only if there are two distinct roots, and negative outside these roots. Here, with a double root, the function touches the x-axis and does not cross it. Thus, there are no positive intervals for $x > 0$. The function is negative for all $x$ outside of $x = -2$.

Therefore, the positive and negative domains are:

$x < 0 : x \ne -2$

$x > 0 :$ none

This matches choice 3 from the list of options.