We are asked to find a quadratic equation in the standard form $ax^2 + bx + c = 0$ with integer coefficients $a$, $b$, and $c$ such that $a$ is positive and $a, b, c$ have no common factor, and the given solutions are $11$ and $-13$.

1. Problem Description

We are asked to find a quadratic equation in the standard form $ax^2 + bx + c = 0$ with integer coefficients $a$, $b$, and $c$ such that $a$ is positive and $a, b, c$ have no common factor, and the given solutions are $11$ and $-13$.

2. Solution Steps

Since the solutions are $x = 11$ and $x = -13$, the factors of the quadratic equation are $(x - 11)$ and $(x - (-13)) = (x + 13)$. Therefore, we can write the quadratic equation as:

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

Expanding the expression, we have:

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

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

The coefficients are $a = 1$, $b = 2$, and $c = -143$. Since $a$ is positive and the greatest common divisor of 1, 2, and -143 is 1, the equation is in the required form.

3. Final Answer

The quadratic equation is $x^2 + 2x - 143 = 0$.