The problem asks us to determine the type of solutions the quadratic equation $2x^2 + 18 = -12x$ has using the discriminant, without solving the equation.

1. Problem Description

The problem asks us to determine the type of solutions the quadratic equation $2x^2 + 18 = -12x$ has using the discriminant, without solving the equation.

2. Solution Steps

First, we rewrite the equation in the standard quadratic form $ax^2 + bx + c = 0$:

$2x^2 + 12x + 18 = 0$

Here, $a = 2$, $b = 12$, and $c = 18$.

The discriminant is given by the formula:

$D = b^2 - 4ac$

Substituting the values of $a$, $b$, and $c$ into the formula, we get:

$D = (12)^2 - 4(2)(18)$

$D = 144 - 144$

$D = 0$

If the discriminant $D > 0$, the equation has two distinct real solutions.

If the discriminant $D = 0$, the equation has one real solution (a repeated root).

If the discriminant $D < 0$, the equation has two nonreal complex solutions.

In our case, $D = 0$, so the quadratic equation has one real solution. Since the coefficients of the quadratic equation are rational, and the discriminant is a perfect square (0), the solution is rational. Therefore, the equation has one rational solution.

3. Final Answer

One rational solution.