Question

If 2x-1/2x=4,find------- i)4x²+1/4x²  ii)8x³-1/8x³
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Answer 1

$2x-1/2x=4 \\ =>(2x-1/2x)^2=16 \\ =>4x^2+1/4x^2=16+2=18$
$(2x-1/2x)^3 \\ =8x^3-1/8x^3-3*2x*1/2x*(2x-1/2x) \\ =8x^3-1/8x^3-3*(2x-1/2x)$
therefore $8x^3-1/8x^3=4^3+3*4=76$

Answer 2

2 x-1/2x=4..................(iii)
(i)4x² + 1/4x² =?
2x-1/2x=4
(2x-1/2x)² = 4²                                               [Squaring on both sides]
(2x)² + (1/2x)² - (2*2x*1/2x) = 16                     [(a-b)² = a² + b² - 2*a*b ]
4x² + 1/4x² - 2 = 16                                       [2x * 1/2x = 1]
4x² + 1/4x² = 16+2
4x² + 1/4x² = 18

(ii)8x³ - 1/8x³ =?
2x - 1/2x = 4
(2x - 1/2x)³ = 4³                                          [Cubing on both sides]
(2x)³ - (1/2x)³ - 3*2x*1/2x(2x - 1/2x) = 64       [(a-b)³ = a³ - b³ - 3*a*b(a-b)]
8x³ - 1/8x³ - 3(2x - 1/2x) = 64                       [2x * 1/2x = 1]
8x³ - 1/8x³ - 3*4 = 64                                 [2x - 1/2x = 4. Also from equation (iii)]
8x³ - 1/8x³ - 12 = 64
8x³ - 1/8x³ = 64+12
8x³ - 1/8x³ = 76