Question

If α, β are the zeroes of the polynomial f(x) = x² + x – 2, then `(α/β - α/β)`.

If α and β are the zeros of the polynomial f(x) = x² + x – 2, find the value of `(1/α - 1/β)`.

Answer 1

By using the relationship between the zeroes of the quadratic polynomial. 

We have  

Sum of zeroes = `(-("Coefficient of"  x))/("Coefficient of"  x^2)` and Product of zeroes = `("Constant term")/("Coefficient of"  x^2)` 

∴ `α + β = (-1)/1` and `αβ = (-2)/1` 

⇒ α + β = –1 and αβ = –2 

Now, `(1/α - 1/β)^2 = ((β - α)/(αβ))^2` 

= `((α + β)^2 - 4αβ)/(αβ)^2`   ...[∵ (β – α)² = (α + β)² – 4αβ] 

= `((-1)^2 - 4(-2))/((-2)^2)`   ...[∵ α + β = –1 and αβ = –2]

= `((-1)^2 - 4(-2))/4` 

= `9/4` 

∵ `(1/α - 1/β)^2 = 9/4` 

⇒ `1/α - 1/β = +-3/2`