If 2a-b=3 and 8a^3-b^3=999, then find the value of 4a^2-b^2.

Question

If 2a-b=3 and 8a^3-b^3=999, then find the value of 4a^2-b^2.

1. A. 61
2. B. 65
3. C. 67
4. D. 63 (Correct answer)

Correct Answer

Option D — 63

Detailed Solution & Explanation

The correct answer is 63.

Key Points

• $(2a-b)^3=8a^3-b^3-6ab(2a-b)\Rightarrow 27=999-6ab(3)\Rightarrow ab=54$.
• $(2a+b)^2=(2a-b)^2+8ab=9+8(54)=441\Rightarrow 2a+b=21$.
• $4a^2-b^2=(2a+b)(2a-b)=21\times3=63$.

Additional Information

• The identity in play is (a − b)³ = a³ − b³ − 3ab(a − b), used here with a = 2a and b = b to extract the product ab.
• The second step uses (a + b)² = (a − b)² + 4ab, which in this form becomes (2a + b)² = (2a − b)² + 8ab.
• The target is a difference of squares, 4a² − b² = (2a + b)(2a − b), so only the sum and difference are needed — never a or b individually.