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

= 13 > 0 

So, the given equation has real roots.

Now, `sqrt(D) = sqrt(13)` 

∴ `α = (-b + sqrt(D))/(2a)`

= `(-(-3) + sqrt(13))/(2 xx 1)`

= `(3 + sqrt(13))/2`

`β = (-b + sqrt(D))/(2a)`

= `(-(-3) + sqrt(13))/(2 xx 1)`

= `(3 + sqrt(13))/2`

= `(3 - sqrt(13))/2` 

Hence, `(3 + sqrt(13))/3` and `(3 - sqrt(13))/3` are the roots of the given equation.