Solve y = -x² - 8x - 20: Finding Where Function Values Are Negative

Look at the following function:

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

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

$f(x) < 0$

To solve this problem, we'll follow these steps:

• Step 1: Identify the discriminant to determine the nature of the roots.
• Step 2: Analyze the quadratic function for signs across its domain.

Now, let's work through each step:

Step 1: Calculate the discriminant $D = b^2 - 4ac$.
For the quadratic $y = -x^2 - 8x - 20$, we have:

• $a = -1$
• $b = -8$
• $c = -20$

The discriminant is $D = (-8)^2 - 4(-1)(-20) = 64 - 80 = -16$. Because $D < 0$, the quadratic has no real roots, meaning it never crosses the x-axis.

Step 2: Since the discriminant is negative and the coefficient $a$ is negative, the parabola opens downward.
With no x-intercepts, the vertex is the point where the maximum value occurs. However, for a downward opening parabola with $D < 0$, it resides above the x-axis for all $x$.

We conclude that the entire parabola remains below the x-axis across all real $x$, meaning the function is negative for all $x$.

Therefore, the solution to the problem is that the function is negative for all $x$.