We are given the quadratic equation $2x(x-1) + 9 = 7$ and asked to determine the nature of its solutions using the discriminant. We do not need to solve for the roots explicitly. The possible solution types are: two rational solutions, two irrational solutions, one rational solution, and two nonreal complex solutions.

1. Problem Description

We are given the quadratic equation $2x(x-1) + 9 = 7$ and asked to determine the nature of its solutions using the discriminant. We do not need to solve for the roots explicitly. The possible solution types are: two rational solutions, two irrational solutions, one rational solution, and two nonreal complex solutions.

2. Solution Steps

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

$2x(x-1) + 9 = 7$

$2x^2 - 2x + 9 = 7$

$2x^2 - 2x + 9 - 7 = 0$

$2x^2 - 2x + 2 = 0$

Now we identify the coefficients: $a = 2$, $b = -2$, and $c = 2$.

The discriminant is given by the formula:

$D = b^2 - 4ac$

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

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

$D = 4 - 16$

$D = -12$

Now we analyze the discriminant:

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

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

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

Since $D = -12 < 0$, the equation has two nonreal complex solutions.

3. Final Answer

Two nonreal complex solutions