Factorize 2y2 - 15xy + 18x2

Analyze the question: We need to factorize the quadratic expression 2y² - 15xy + 18x², which is a quadratic in y with x as a constant multiplier.

Concept: For a quadratic ay² + bxy + cx², find two numbers that multiply to a×c and add to b. Then split the middle term and factorize by grouping.

Solution:

• a × c = 2 × 18 = 36
• We need two numbers that multiply to +36 and add to -15. These are -3 and -12.

Split the middle term:

2y² - 3xy - 12xy + 18x²

Group:

= y(2y - 3x) - 6x(2y - 3x)

= (2y - 3x)(y - 6x)

Check by expanding option B:

(2y - 3x)(y - 6x) = 2y² - 12xy - 3xy + 18x² = 2y² - 15xy + 18x² ✓

Why the others fail:

• A: (2y - 3x)(y + 6x) gives 2y² + 9xy - 18x², which is wrong.
• C: (2y + 3x)(y - 6x) gives 2y² - 9xy - 18x², which is wrong.
• D: (3y + 2x)(y - 6x) gives 3y² - 16xy - 12x², which is wrong.

Tip: When the last term is positive and the middle term is negative, both brackets carry negative signs. Eliminate options with mixed signs immediately (A, C, D), which saves time in the exam. Always verify by quickly expanding.