If x^2 - 5x + 1 = 0, what is the value of x^3 + 2(x + 1/x + 1/x^3) ?

Question

If  x^2-5x+1=0 , what is the value of  x^3+2(x+\frac{1}{x})+\frac{1}{x^3 }  ?

A.  90 
 
B.  100 
 
C.  110 
 
D.  120 
 
E.  130

Aabhash777 · Answer

x^2 - 5x + 1 = 0

Solution

x^2 - 5x + 1 = 0
x(x-5+ 1/x) = 0
x=0 (Discard, not needed)
or 
x + 1/x =5 .........................(A)

Now
Cubing on both sides
(x + 1/x)^3 =125
x^3+3(x+ 1/x) + 1/x^3 = 125
x^3+1/x^3 = 125 - 3*5 = 110 (From A)

Now,

x^3 + 2(x + 1/x) + 1/x^3 
= 110+ 2*5 = 120

Qorda · Answer

is there easier solution to this question?

hr1212 · Answer

x^2-5x+1=0

Dividing whole equation by x, gives us

x - 5 + \frac{1}{x} = 0 
 x + \frac{1}{x} = 5  --- (1)

Squaring this gives us,

x^2 + 2 + \frac{1}{x^2} = 25 
 x^2 + \frac{1}{x^2} = 23  --- (2)

Recall the cube formula, would come in handy,

x^3 + \frac{1}{x^3} = (x + \frac{1}{x})(x^2 - 1 + \frac{1}{x^2}) 
 x^3 + \frac{1}{x^3} = 5*22  
 x^3 + \frac{1}{x^3} = 110

We are asked to calculate value for -

x^3+2(x+\frac{1}{x})+\frac{1}{x^3 }  
110 + 2*5
110 + 10
120

Answer: D

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