Factorize completely: 27x$^2$ - 48y$^2$

Analyze the question: We need to factorize 27x² - 48y² completely. That means pulling out any common factor first, then checking whether what remains can be factorized further.

Concept: Two key ideas are used:

1. Take out the highest common factor (HCF).
2. Difference of two squares: a² - b² = (a + b)(a - b)

Solution:

Step 1: Find the HCF of 27 and 48. Both are divisible by 3, so the HCF is 3.

27x² - 48y² = 3(9x² - 16y²)

Step 2: Recognize the difference of two squares.

• 9x² = (3x)²
• 16y² = (4y)²

So 9x² - 16y² = (3x)² - (4y)² = (3x + 4y)(3x - 4y)

Step 3: Combine.

27x² - 48y² = 3(3x + 4y)(3x - 4y)

Check by expanding: 3(3x + 4y)(3x - 4y) = 3(9x² - 16y²) = 27x² - 48y² ✓

Why the other options are wrong:

• B and D are perfect squares like (a + b)² or (a - b)², which would give a middle term, so they cannot equal a difference of squares.
• C: 3(9x - 16y)(9x + 16y) = 3(81x² - 256y²), which is not the original expression.

Answer: A

Exam tip: Always look for a common factor first, then check for a difference of two squares. If you see two terms with a minus sign between them and both are perfect squares (after taking out the HCF), think (a + b)(a - b).