Solve the Quadratic Function Inequality: When is y = -0.9x² Less Than Zero?

Given the function:

$y=-0.9x^2$

Determine for which values of x the following holds:

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

To solve this problem, we need to determine when the quadratic function $y = -0.9x^2$ is less than zero. This requires analyzing the entire set of x-values.

• Given that the function $y = -0.9x^2$ is a parabola opening downward because $a = -0.9 < 0$, the function will take negative values for all $x$ except when $x^2 = 0$.
• The vertex of this parabola is at the origin, $x = 0$, since terms for linear ($bx$) and constant ($c$) are zero in this function.
• At $x = 0$, $y = -0.9(0)^2 = 0$, meaning at $x = 0$, the function is exactly zero.
• For any $x \neq 0$, the term $x^2 > 0$, hence $-0.9x^2 < 0$, indicating the function is negative.

Therefore, the function $y = -0.9x^2$ is less than zero for all values except at $x = 0$.

Consequently, the solution to the problem is that the function is negative for all $x \ne 0$.

This corresponds to choice: $x\ne0$.