If 2a - b = 3 and 8a³ - b³ = 999, then find the value of 4a² - b².

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Given:
2a - b = 3  ...(1)
8a³ - b³ = 999 ...(2)

Step 1: Recognize that 8a³ = (2a)³, so equation (2) can be written as:
(2a)³ - b³ = 999

Step 2: Use the identity for difference of cubes:
(x³ - y³) = (x - y)(x² + xy + y²)
Here, x = 2a and y = b.

So,
(2a - b)((2a)² + 2a·b + b²) = 999

Step 3: From equation (1), 2a - b = 3.
Substitute into the above:
3 * ((2a)² + 2a·b + b²) = 999

Step 4: Simplify:
(2a)² + 2a·b + b² = 999 / 3 = 333

Step 5: Calculate (2a)² = 4a², so:
4a² + 2ab + b² = 333 ...(3)

Step 6: We need to find 4a² - b².
From (3), rearranged:
4a² - b² = 333 - 2ab ...(4)

Step 7: From equation (1), b = 2a - 3.
Calculate ab:
ab = a * b = a * (2a - 3) = 2a² - 3a

Step 8: Substitute ab into (4):
4a² - b² = 333 - 2(2a² - 3a) = 333 - 4a² + 6a

Step 9: Rearranged:
4a² - b² + 4a² = 333 + 6a
8a² - b² = 333 + 6a

Step 10: We need to find a numeric value, so find a and b.
From (1): b = 2a - 3
From (2): 8a³ - b³ = 999
Substitute b:
8a³ - (2a - 3)³ = 999

Step 11: Expand (2a - 3)³:
(2a - 3)³ = 8a³ - 36a² + 54a - 27

Step 12: Substitute back:
8a³ - [8a³ - 36a² + 54a - 27] = 999
8a³ - 8a³ + 36a² - 54a + 27 = 999
36a² - 54a + 27 = 999

Step 13: Simplify:
36a² - 54a + 27 - 999 = 0
36a² - 54a - 972 = 0
Divide entire equation by 18:
2a² - 3a - 54 = 0

Step 14: Solve quadratic:
2a² - 3a - 54 = 0
Using quadratic formula:
a = [3 ± sqrt(9 + 432)] / 4
= [3 ± sqrt(441)] / 4
= [3 ± 21] / 4

Step 15: Two roots:
a = (3 + 21)/4 = 24/4 = 6
or
a = (3 - 21)/4 = -18/4 = -4.5

Step 16: Find corresponding b:
If a = 6, b = 2*6 - 3 = 12 - 3 = 9
If a = -4.5, b = 2*(-4.5) - 3 = -9 - 3 = -12

Step 17: Calculate 4a² - b² for both:
For a=6, b=9:
4a² - b² = 4*(36) - 81 = 144 - 81 = 63
For a=-4.5, b=-12:
4a² - b² = 4*(20.25) - 144 = 81 - 144 = -63 (not matching options)

Step 18: Valid answer is 63.

Hence, the value of 4a² - b² is 63.

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