The given expression is \( x^2 - 27xy + 72y^2 \). To solve it, we will factorize the quadratic expression.

### Step 1: Identify the quadratic form
The expression is in the form \( ax^2 + bxy + cy^2 \), where:
- \( a = 1 \)
- \( b = -27 \)
- \( c = 72 \)

### Step 2: Find the factors of the quadratic expression
We need to find two binomials of the form \( (x - py)(x - qy) \) such that:
\[
x^2 - (p + q)xy + pqy^2 = x^2 - 27xy + 72y^2
\]
Here, \( p + q = 27 \) and \( pq = 72 \).

### Step 3: Solve for \( p \) and \( q \)
We look for factors of 72 that add up to 27:
- The pair \( (p, q) \) that satisfies \( p + q = 27 \) and \( pq = 72 \) is \( p = 3 \) and \( q = 24 \).

### Step 4: Write the factored form
Thus, the expression can be factored as:
\[
x^2 - 27xy + 72y^2 = (x - 3y)(x - 24y)
\]

### Final Answer:
\[
x^2 - 27xy + 72y^2 = (x - 3y)(x - 24y)
\]

Would you like further details or have any questions?

Here are some related questions you might find interesting:
1. How would you factor a similar expression like \( x^2 + 15xy + 36y^2 \)?
2. What is the discriminant of the quadratic expression \( x^2 - 27xy + 72y^2 \)?
3. How does the factoring process change if the coefficients are not integers?
4. Can the expression \( x^2 - 27xy + 72y^2 \) be factored differently if using complex numbers?
5. How would you solve the equation \( x^2 - 27xy + 72y^2 = 0 \) for \( x \) in terms of \( y \)?

**Tip:** Always check if you can factor out a common factor before proceeding with more complex factoring techniques.

Learn how to factor the quadratic expression x² - 27xy + 72y² with a detailed step-by-step solution. This algebra problem covers key concepts like factoring binomials and the relationship between the sum and product of roots. Suitable for students in Grades 8-10.

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