The problem asks to write a quadratic equation in the standard form $ax^2 + bx + c = 0$ with the given solutions $x = 3$ and $x = 10$. Also, $a$ must be positive, and $a$, $b$, and $c$ must be integers with no common factors.

1. Problem Description

The problem asks to write a quadratic equation in the standard form $ax^2 + bx + c = 0$ with the given solutions $x = 3$ and $x = 10$. Also, $a$ must be positive, and $a$, $b$, and $c$ must be integers with no common factors.

2. Solution Steps

Since $x=3$ and $x=10$ are solutions to the quadratic equation, we can write the equation in factored form as:

$(x - 3)(x - 10) = 0$

Expanding this expression, we get:

$x^2 - 10x - 3x + 30 = 0$

Combining like terms, we have:

$x^2 - 13x + 30 = 0$

Here, $a=1$, $b=-13$, and $c=30$. Since $a$ is positive and the coefficients have no common factors, this is the desired quadratic equation.

3. Final Answer

$x^2 - 13x + 30 = 0$