Question

x^2+1/x^2=51 then find x^3-1/x^3

Answer 1

(x^2)+1/(x^2)=51

we know (a-b)^2=a^2 + b^2 -2ab
Applying (a-b)^2 because the answer is needed for (x^3)-1/(x^3) for which we will need value of x-1/x

Now, subtracting 2(x)(1/x) from both sides
(x^2)+[1/(x^2)] -2(x)(1/x)=51- 2(x)(1/x)

(x-1/x)^2=51-2=49
Thus x-1/x = 7 or -7

(x-1/x)^3=(x^3)-1/(x^3)- 3(x)(1/x)(x-1/x)
=(x^3)-1/(x^3)-3(x-1/x)


applying x-1/x =7
7^3=(x^3)-1/(x^3)-3(7)
(x^3)-1/(x^3)=343+21=364

Applying x-1/x =-7
-7^3=(x^3)-1/(x^3)-3(-7)
(x^3)-1/(x^3)=-343-21=-364

Hope this helps.
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