Solve the following mathematical problems:

What should be the values of a and b for which $64 x^{2} - 9 a x^{2} + 108 x - b$ will be a perfect cube?

Created: 1 year ago | Updated: 1 year ago

Updated: 1 year ago

$w e k n o w, {(x - y)}^{3} = {(x)}^{3} - 3 x^{2} y + 3 x y^{2} - y^{3} N o w, 64 x^{3} - 9 a x^{2} + 108 x - b = {(4 x)}^{3} - 3 \times {(4 x)}^{2} \times 3 + 3 \times 4 x \times {(3)}^{2} - {(3)}^{3} = {(4 x - 3)}^{3} I n h e r e, i n t h e 2^{n d} t e r m 3 \times {(4 x)}^{2} \times 3 = 3 \times {(4)}^{2} \times x^{2} \times 3 S o, i n h e r e 4^{2} = 16 = a I n t h e 3^{r d} t e r m = 3 \times 4 x \times 3^{2} [3 x y^{2}] A n d i n 4^{t h} t e r r m = 3^{3} = b S o, b = 3^{3} = 27 a n s . a = 16 & b = 27$