Identify When y = -x² - 2x - 3 Surpasses Zero: Solving the Quadratic Inequality

Look at the following function:

$y=-x^2-2x-3$

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

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

To solve this problem, follow these steps:

• Step 1: Find the roots of the function using the quadratic formula. The equation to solve is $-x^2 - 2x - 3 = 0$.
• Step 2: The quadratic formula gives $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. For our function, $a = -1$, $b = -2$, and $c = -3$.
• Step 3: Calculate the discriminant: $b^2 - 4ac = (-2)^2 - 4(-1)(-3) = 4 - 12 = -8$.
• Step 4: Since the discriminant is negative, there are no real roots. Therefore, $f(x)$ never crosses the x-axis and has no real x-intercepts.
• Step 5: Understand the graph behavior: since the parabola opens downward and the vertex is its maximum point without crossing the x-axis, the function is negative for all real $x$.

Therefore, the solution to the problem is that the function has no positive domain.