Question

If a and 3 are the zeros of the quadratic polynomial f(x) = x² + x − 2, find the value of `1/alpha-1/beta`.

Answer 1

Since 𝛼 and 𝛽 are the roots of the polynomial x + x – 2

∴ Sum of roots α + β = 1

Product of roots αβ 2 ⇒ `-1/beta`

`=(beta-alpha)/alphabeta*(alpha-beta)/alphabeta`

`=(sqrt((alpha+beta)^2-4alphabeta))/(alphabeta)`

`=sqrt(1+8)/(+2)`

`=3/2`

Answer 2

Given if α and β​​​​​ are the solutions of the polynomial f(x) = x2 + x − 2.

So, first let us find zeros of f(x) = 0:

The middle term x is expressed as sum of 2x and −x such that its product is equals to product of extreme terms. 

(-2) x x2 = -2x2

Thus, x2 + 2x - x - 2 = 0

x(x + 2) - 1(x + 2) = 0

(x + 2)(x - 1) = 0

(x + 2) = 0 or (x - 1) = 0

=> x = -2 or x = 1

∴ α, β = (1, -2) or (-2, 1)

Case 1: When (α, β) = (1, -2)

`(1/alpha - 1/beta) = 1/1 - 1/(-2)`

= `1 + 1/2`

= `(2 + 1)/2`

∴ `1/alpha - 1/beta = (-3)/2`

Hence, `1/alpha - 1/beta = (-3)/2 or 3/2`