Solve Quadratic Inequality: When is -3x² - 9 Less Than Zero

Given the function:

$y=-3x^2-9$

Determine for which values of x the following holds:

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

Given the quadratic function $f(x) = -3x^2 - 9$, we want to determine when $f(x) < 0$.

First, observe that the function is a downward-opening parabola because the coefficient of $x^2$ is negative ($-3$). This means the parabola opens downwards.

To find when the parabola is below the x-axis ($f(x) < 0$), we should first check whether there are any real roots, since this implies crossing the x-axis.

The function $f(x) = -3x^2 - 9$ is reformulated as:

$0 = -3x^2 - 9$.

To find the roots, rearrange and solve:

$-3x^2 = 9$ or $x^2 = -3$.

Since $x^2 = -3$ yields no real solutions (as no real number squared equals a negative), there are no x-intercepts.

This indicates the parabola does not cross the x-axis and is entirely below it (since it opens downward and has no real roots).

Thus, the function $f(x) < 0$ for all real values of $x$.

Therefore, the condition $f(x) < 0$ is satisfied for all x, meaning the correct choice is:

All x