Solve the Quadratic Inequality: 2x² - 12x + 18 < 0

To solve this inequality $2x^2 - 12x + 18 < 0$, we perform the following steps:

• Step 1: Compute the discriminant. Here, $a = 2$, $b = -12$, and $c = 18$. Plug these into the discriminant formula:

$\Delta = b^2 - 4ac = (-12)^2 - 4 \times 2 \times 18 = 144 - 144 = 0$

• Step 2: Interpret $\Delta = 0$. A discriminant of zero means the quadratic equation has one repeated real root. The graph of the quadratic is a parabola that touches the x-axis at a single point, at the vertex.
• Step 3: Since the parabola described by the quadratic has no interval crossing the x-axis (it only touches it at a point), the inequality $2x^2 - 12x + 18 < 0$ has no solutions. A parabola touching the x-axis at a single vertex point does not lie below the x-axis.

Therefore, there are no values of $x$ for which the quadratic expression $2x^2 - 12x + 18 < 0$. The function does not attain negative values.

The correct answer to the multiple-choice question is therefore:
The function is negative for all values of x.