Question

Find the roots of the following equation, if they exist, by applying the quadratic formula:

`x + 1/x = 3, x ≠ 0`

Answer 1

The given equation is 

`x + 1/x = 3, x ≠ 0` 

⇒ `(x^2 + 1)/x = 3 ` 

⇒ x² + 1 = 3x 

⇒ x² – 3x + 1 = 0 

This equation is of the form ax² + bx + c = 0, where, a = 1, b = –3 and c = 1. 

∴ Discriminant, D = b² – 4ac

= (–3)² – 4 × 1 × 1

= 9 – 4

= 5 > 0 

So, the given equation has real roots.

Now, `sqrt(D) = sqrt(5)` 

∴ `a = (-b + sqrt(D))/(2a)`

= `(-(-3) + sqrt(5))/(2 xx 1)`

= `(3 + sqrt(5))/2` 

`β = (-b - sqrt(D))/(2a)`

= `(-(-3) - sqrt(5))/(2 xx 1)`

= `(3 - sqrt(5))/2`  

Hence, `(3 + sqrt(5))/2` and `(3 - sqrt(5))/2` are the roots of the given equation.