We are given the quadratic equation $2x^2 + 18 = -12x$ and we need to determine the type of solutions it has by using the discriminant. We do not need to solve the equation. The possible answer choices are: two rational solutions, two irrational solutions, one rational solution, two nonreal complex solutions.

1. Problem Description

We are given the quadratic equation $2x^2 + 18 = -12x$ and we need to determine the type of solutions it has by using the discriminant. We do not need to solve the equation. The possible answer choices are: two rational solutions, two irrational solutions, one rational solution, two nonreal complex solutions.

2. Solution Steps

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

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

Now, identify the coefficients $a$, $b$, and $c$.

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

The discriminant is given by the formula:

$D = b^2 - 4ac$

Substitute the values of $a$, $b$, and $c$ into the discriminant formula:

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

$D = 144 - 144$

$D = 0$

Now, determine the type of solutions based on the value of the discriminant:

- If $D > 0$ and $D$ is a perfect square, there are two distinct rational solutions.

- If $D > 0$ and $D$ is not a perfect square, there are two distinct irrational solutions.

- If $D = 0$, there is one rational solution (a repeated root).

- If $D < 0$, there are two nonreal complex solutions.

Since $D = 0$, there is one rational solution.

3. Final Answer

One rational solution