The problem asks us to determine the nature of the roots of the quadratic equation $x^2 - 7x + 3 = 0$.

1. Problem Description

The problem asks us to determine the nature of the roots of the quadratic equation $x^2 - 7x + 3 = 0$.

2. Solution Steps

To determine the nature of the roots, we need to calculate the discriminant, which is given by the formula:

$D = b^2 - 4ac$, where $a$, $b$, and $c$ are the coefficients of the quadratic equation $ax^2 + bx + c = 0$.

In our equation, $x^2 - 7x + 3 = 0$, we have $a = 1$, $b = -7$, and $c = 3$.

Now, we can calculate the discriminant:

$D = (-7)^2 - 4(1)(3)$

$D = 49 - 12$

$D = 37$

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

Since $D = 37$ is not a perfect square, the roots are irrational.

3. Final Answer

The roots are real, irrational, and unequal.