MathematicsIf x3+1×3=110, then what is the value of x+1x?

If $x^{3} + \frac{1}{x^{3}} = 110$ , then what is the value of $x + \frac{1}{x}$ ?

A
5
B
10
C
15
D
None

Solution:

= a^{3} + b^{3} + 3 ab (a + b)

{(x + \frac{1}{x})}^{3} = x^{3} + {(\frac{1}{x})}^{3} + 3 \times x \times \frac{1}{x} (x + \frac{1}{x}) = x^{3} + \frac{1}{x^{3}} + 3 (x + \frac{1}{x})

Let y =

(x + \frac{1}{x})

\Rightarrow y^{3} = 110 + 3 y

\Rightarrow y^{3} - 3 y - 110 = 0

\Rightarrow y^{3} - 5 y^{2} + 5 y^{2} - 3 y - 110 = 0

\Rightarrow y^{2} (y - 5) + 5 y^{2} - 25 y + 22 y - 110 = 0

\Rightarrow y^{2} (y - 5) + 5 y^{} (y - 5) + 22 y (y - 5) = 0

\Rightarrow

(y - 5) {(y}^{2} + 5 y^{} + 22 y

) = 0

\Rightarrow y = 5

or y =

\frac{- 5 \pm 3 \sqrt{7} i}{2}

\Rightarrow x + \frac{1}{x} = 5

Hence, Option (1) is the correct answer.

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