Question

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

`1/x - 1/(x - 2) = 3, x ≠ 0, 2`

Answer 1

The given equation is 

`1/x - 1/(x - 2) = 3, x ≠ 0, 2`

⇒ `(x - 2 - x)/(x(x - 2)) = 3` 

⇒ `-2/(x^2 - 2x) = 3` 

⇒ –2 = 3x² – 6x 

⇒ 3x² – 6x + 2 = 0 

This equation is of the form ax² + bx + c = 0 Where a = 3, b = –6 and c = 2. 

∴ Discriminant, D = b² – 4ac

= (–6)² – 4 × 3 × 2

= 36 – 24

= 12 > 0 

So, the given equation has real roots.

Now, `sqrt(D) = sqrt(12) = 2sqrt(3)` 

∴ `α = (-b + sqrt(D))/(2a)`

= `(-(-6) + 2sqrt(3))/(2 xx 3)`

= `(6 + 2sqrt(3))/6`

= `(3 + sqrt(3))/3` 

`β = (-b - sqrt(D))/(2a)`

= `(-(-6)-2sqrt(3))/(2 xx 3)`

= `(6 - 2sqrt(3))/6`

= `(3 - sqrt(3))/3`  

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