The problem asks to write a quadratic equation in standard form $ax^2 + bx + c = 0$ given the solutions $9$ and $2$. The coefficient $a$ must be positive and $a$, $b$, and $c$ must have no common factors.

1. Problem Description

The problem asks to write a quadratic equation in standard form $ax^2 + bx + c = 0$ given the solutions $9$ and $2$. The coefficient $a$ must be positive and $a$, $b$, and $c$ must have no common factors.

2. Solution Steps

Since $9$ and $2$ are the solutions, we can write the equation in factored form as $(x - 9)(x - 2) = 0$.

Expanding the expression, we get:

$x^2 - 2x - 9x + 18 = 0$

$x^2 - 11x + 18 = 0$

This equation is in the standard form $ax^2 + bx + c = 0$, with $a = 1$, $b = -11$, and $c = 18$.

Since $a = 1$ is positive, and $1$, $-11$, and $18$ have no common factors, the equation is in the required form.

3. Final Answer

$x^2 - 11x + 18 = 0$