We are asked to determine the type of solutions of the quadratic equation $36x^2 - 60x + 25 = 0$ using the discriminant.

1. Problem Description

We are asked to determine the type of solutions of the quadratic equation $36x^2 - 60x + 25 = 0$ using the discriminant.

2. Solution Steps

The general form of a quadratic equation is $ax^2 + bx + c = 0$. In our case, $a = 36$, $b = -60$, and $c = 25$. The discriminant, denoted by $\Delta$, is given by the formula:

$\Delta = b^2 - 4ac$

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

$\Delta = (-60)^2 - 4(36)(25)$

$\Delta = 3600 - 3600$

$\Delta = 0$

Now, we analyze the value of the discriminant to determine the nature of the roots:

- If $\Delta > 0$, the quadratic equation has two distinct real roots.

- If $\Delta = 0$, the quadratic equation has exactly one real root (a repeated root).

- If $\Delta < 0$, the quadratic equation has two complex (non-real) roots.

Since $\Delta = 0$, the given quadratic equation has exactly one real root.

Furthermore, since $a$, $b$, and $c$ are rational numbers, the single root will be rational.

3. Final Answer

One rational solution.