Question

The number of real roots of

( x + 1/x )³ + ( x + 1/x ) = 0 is ?

explain✏✏✏

Answer 1

Let x + 1/x = y.
Since the minimum value of y is 2 for a real x, y cannot be 0.

Given y^3 + y = 0
=> y×(y^2 +1) = 0
=> y^2 +1 = 0
=> y = + i or -i

So there are no real roots for the given equation in x.

Let us see why y = x +1/x not equal to 0.
x + 1/x = 0
=> x^2 + 1 = x × 0 = 0
=> x = +i or -i.
So for a real x , x+1/x is not 0.

Answer 2

Let x + 1/x = y.
Since the minimum value of y is 2 for a real x, y cannot be 0.

Given y^3 + y = 0
=> y×(y^2 +1) = 0
=> y^2 +1 = 0
=> y = + i or -i

So there are no real roots for the given equation in x.

Let us see why y = x +1/x not equal to 0.
x + 1/x = 0
=> x^2 + 1 = x × 0 = 0
=> x = +i or -i.
So for a real x , x+1/x is not 0.