Question

x3+1/x3=18 find the value of x+1/x
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Answer 1

X3+1/X3=18

X3+X3=18/1=18

6x=18

x=18/6

x=3

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Answer 2

Step-by-step explanation:

x3+1x3=18

Observe that a3+b3=(a+b)(a2−ab+b2)=(a+b)((a+b)2−3ab); upon taking a=x3 and b=1x3 we arrive at

(x+1x)[(x+1x)2−3]=18

Hence if t=x+1x then t(t2−3)=18.

Moreover, t3−3t−18=0=(t−3)(t2+3t+6);

hence t=3 or t=−3±i15√2.

Now we use the fact that t=x+1x to get:

t2=(x+1x)2=x2+2+1x2

x2−2+1x2=t2−4

(x−1x)2=t2−4

x−1x=±t2−4−−−−−√

as desired.

If t=3 then x−1x=±5–√.

If t=−3+i15√2, then x−1x=±−11−3i15√2−−−−−−−−√=±5√−3i3√2.

Similarly, if t=−3−i15√2, then x−1x=±5√+3i3√2.