Question

If x^3+1/x^3 =110 find x+1/x​

Answer 1

Solution:

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

Let $y = x +\frac{1}{x}$

⇒$y^{3} = 110 + 3y$

⇒$y^{3} - 3y - 110 = 0$

⇒$y^{3} - 5y^{2} + 5y^{2} - 3y - 110 = 0$

⇒$y^{2} (y - 5) + 5y^{2} -25y +22y - 110 = 0$

⇒$y^{2} (y - 5) + 5y (y -5 ) + 22y(y-5) = 0$

⇒$(y-5) (y^{2} +5y +22) = 0$

⇒$y = 5 or y = \frac{-5 ± 3\sqrt{7}i }{2}$

∴$x + \frac{1}{x} = 5$

Hope it helps you dear !!!