Solve y=-x²-8x-20: Finding Where Function is Positive

Look at the following function:

$y=-x^2-8x-20$

Determine for which values of $x$ the following is true:

$f\left(x\right)>0$

To find where the function $y = -x^2 - 8x - 20$ is positive, follow these steps:

Step 1: Calculate the discriminant of the quadratic function.

• The quadratic discriminant is $\Delta = b^2 - 4ac$.
• In our case, $a = -1$, $b = -8$, $c = -20$, so:
• $\Delta = (-8)^2 - 4(-1)(-20) = 64 - 80 = -16$.

Step 2: Analyze the discriminant.

• Since $\Delta < 0$, the quadratic equation has no real roots.
• The parabola does not cross the x-axis, and opens downward because $a = -1 < 0$.

Step 3: Determine the sign of the function across the real domain.

• Without real roots and a parabola opening downwards, the function is negative across its entire domain.

Given this analysis, there are no values of $x$ for which the function $f(x) > 0$. Therefore, the function has no positive domain.

Conclusion: The function has no positive domain.