If 2x + 3y = 4 and 4x2 + 9y2 = 64, then what is the value of 8x3 + 27y3.
Show answer & explanation
Correct answer: C
Explanation
The correct answer is option 3 i.e. 352.
Given that
If 2x + 3y = 4 and 4x2 + 9y2 = 64,
Formula used
a3 + b3 = (a + b) (a2 – ab + b2)
(a + b)2 = a2 + b2 + 2ab
Calculation
a = 2x, b = 3y,
⇒ (2x + 3y)2 = 4x2 + 9y2 + 12xy
⇒ 12xy = 42 - 64 = 16 - 64 = - 48
⇒ xy = -48/12 = -4
Now,
⇒ (2x)3 + (3y)3 = (2x + 3y) [(2x)2 - 2x × 3y + (3y)2]
⇒ 8x3 + 27y3 = 4 × [4x2 - 6xy + 9y2]
⇒ 8x3 + 27y3 = 4 × [64 - (6 × -4)]
⇒ 8x3 + 27y3 = 4 × [64 + 24] = 4 × 88 = 352