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Question

If sum of values of x of equation x23+x132=0 is α . Then find |α|.

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Solution

x23+x132=0
In this case we can see that,
23=2(13)
and so one of the exponents is twice the other so it looks like we’ve got an equation that is reducible to quadratic in form. The substitution will then be
u=x13
u2=x23
Substituting this into the equation gives,
u2+u2=0
By using formula b±b24ac2a we get factors,
u=1+3 and u=13
So, x13=1+3
x=(1+3)3=(1)3+(3)3+3(1)(3)(1+3)
x=1+3333(1+3)=10+63 ----- ( 1 )
Now, x13=(13)
x=(13)3=(1)3(3)33(1)(3)(13)
x=1063 ------ ( 2 )
Sum of values of x =(10+63)+(1063) [ From ( 1 ) and ( 2 ) ]
α=20
|α|=|20|=20

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