We are given the quadratic equation $4x(x-1) + 17 = 14$. We need to determine the type of solutions this equation has using the discriminant. We are asked not to solve the equation explicitly.

1. Problem Description

We are given the quadratic equation $4x(x-1) + 17 = 14$. We need to determine the type of solutions this equation has using the discriminant. We are asked not to solve the equation explicitly.

2. Solution Steps

First, simplify the equation:

$4x(x-1) + 17 = 14$

$4x^2 - 4x + 17 = 14$

$4x^2 - 4x + 17 - 14 = 0$

$4x^2 - 4x + 3 = 0$

Now, we have a quadratic equation in the form $ax^2 + bx + c = 0$, where $a = 4$, $b = -4$, and $c = 3$.

The discriminant, denoted by $\Delta$, is given by the formula:

$\Delta = b^2 - 4ac$

Plug in the values of $a$, $b$, and $c$:

$\Delta = (-4)^2 - 4(4)(3)$

$\Delta = 16 - 48$

$\Delta = -32$

Since the discriminant is negative ($\Delta < 0$), the quadratic equation has two nonreal complex solutions (complex conjugate roots).

3. Final Answer

Two nonreal complex solutions