The problem asks us to write a quadratic equation in the standard form $ax^2 + bx + c = 0$, given the solutions $x = 11$ and $x = -13$. The coefficients $a$, $b$, and $c$ must be integers with no common factors, and $a$ must be positive.

1. Problem Description

The problem asks us to write a quadratic equation in the standard form $ax^2 + bx + c = 0$, given the solutions $x = 11$ and $x = -13$. The coefficients $a$, $b$, and $c$ must be integers with no common factors, and $a$ must be positive.

2. Solution Steps

If $x = 11$ is a solution, then $(x - 11)$ is a factor of the quadratic equation. Similarly, if $x = -13$ is a solution, then $(x + 13)$ is a factor of the quadratic equation. Therefore, the quadratic equation can be written as:

$(x - 11)(x + 13) = 0$

Expanding the expression, we have:

$x^2 + 13x - 11x - 11 \cdot 13 = 0$

$x^2 + 2x - 143 = 0$

In this case, $a=1$, $b=2$, and $c=-143$. Since the greatest common divisor of 1, 2, and -143 is 1, the coefficients have no common factors.

So, the quadratic equation is $x^2 + 2x - 143 = 0$.

3. Final Answer

$x^2 + 2x - 143 = 0$