Question

X^2+1/4x^2=8.
Find X^3+1/8x^3

Answer 1

Answer:

We have x² + 1/(4x²) = 8.

But x² + 1/(4x²) = (x + 1/2x)² - 2*x*1/2x => (x + 1/2x)² -1 = 8

=> (x + 1/2x)² = 9

=> x + 1/2x = ±3

In both cases obtained value of x is real.

Now let S = x³ + 1/8x³ = (x + 1/2x)³ - 3*(x)*(1/2x)*(x+1/2x) = (x + 1/2x)³ - (3/2)*(x+1/2x)

S = (+3)³ - (3/2)(+3) or S = (-3)³ - (3/2)(-3)

=> S = 27 - 9/2 or -27+ 9/2

=> S = ± 22.5

So x³ + 1/8x³ = ± 22.5

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