The problem asks us to find a quadratic equation in the form $ax^2 + bx + c = 0$ given its solutions $x = 12$ and $x = 13$. The coefficients $a$, $b$, and $c$ must be integers with no common factor, and $a$ must be positive.

1. Problem Description

The problem asks us to find a quadratic equation in the form $ax^2 + bx + c = 0$ given its solutions $x = 12$ and $x = 13$. The coefficients $a$, $b$, and $c$ must be integers with no common factor, and $a$ must be positive.

2. Solution Steps

Since $x = 12$ and $x = 13$ are the solutions of the quadratic equation, we can write the equation in factored form as $(x - 12)(x - 13) = 0$.

Expanding this, we get:

$x^2 - 13x - 12x + (12)(13) = 0$

$x^2 - 25x + 156 = 0$

Here, $a = 1$, $b = -25$, and $c = 156$. Since the greatest common divisor of 1, -25, and 156 is 1, the condition of no common factor is met. Also, $a = 1$ which is positive.

3. Final Answer

The quadratic equation is $x^2 - 25x + 156 = 0$.