Solve the Inequality: When -3x² + 6x - 9 is Less Than Zero

To determine for which values of $x$ the function $f(x) = -3x^2 + 6x - 9 < 0$, follow these steps:

• Step 1: Identify the coefficient information. The function is $-3x^2 + 6x - 9$ with $a = -3$, $b = 6$, $c = -9$.
• Step 2: Determine the orientation of the parabola. The leading coefficient $a = -3$ is negative, so the parabola opens downward.
• Step 3: Calculate the vertex using the formula $x = -\frac{b}{2a} = -\frac{6}{2 \times (-3)} = 1$. The vertex is at $x = 1$.
• Step 4: Find the function's value at the vertex: $f(1) = -3(1)^2 + 6(1) - 9 = -3 + 6 - 9 = -6$. The function is negative at the vertex.
• Step 5: Calculate the discriminant $\Delta = b^2 - 4ac = 6^2 - 4(-3)(-9) = 36 - 108 = -72$. The discriminant is negative, indicating no real roots.

Since the quadratic opens downward and does not cross or touch the x-axis, it remains entirely below the x-axis for all values of $x$. Therefore, the function is negative for all $x$.

Thus, the solution is: The function is negative for all $x$.