Factoring quadratic expressions requires certain conditions to ensure that the expression can be written as a product of two binomials. The conditions depend on the form and coefficients of the quadratic expression. Let's break down these conditions.

1. Standard Form:

A quadratic expression must be in the form: $ax^2 + bx + c$ where $a$, $b$, and $c$ are constants. For quadratics involving two variables (like the one we just worked on), the general form is: $ax^2 + bxy + cy^2$

2. Integer Coefficients:

Factoring over the integers is possible if the expression can be split into two binomials, each with integer coefficients. This is usually applicable when:

• The discriminant ($\Delta = b^2 - 4ac$) is a perfect square, ensuring that the roots of the quadratic equation are rational.

3. Trial and Error with Factors:

For simpler quadratics, factoring involves finding two numbers $p$ and $q$ such that: $p \cdot q = ac \quad \text{and} \quad p + q = b$ This helps in breaking down the quadratic into simpler terms.

4. Special Patterns:

Some quadratics follow recognizable patterns that make them easy to factor:

• Difference of Squares: $a^2 - b^2 = (a - b)(a + b)$
• Perfect Square Trinomials: $a^2 + 2ab + b^2 = (a + b)^2$
• Quadratics in Two Variables: As in the case of the expression $x^2 - 11xy + 18y^2$, this can be factored into two binomials involving both $x$ and $y$.

5. Factorization over Non-Integers:

If an expression cannot be factored over the integers, you may still factor it using non-integer values (e.g., fractions or radicals). In these cases, solving for the roots via the quadratic formula might be necessary first: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ This method is useful when the discriminant is not a perfect square.

6. Factoring by Grouping:

For some quadratic expressions, especially those with more than one variable, factoring by grouping can be a useful method. This involves splitting the middle term in such a way that you can factor two groups