Question

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

Answer 1

Given: α and β are the zeros of f(x) = x² + x – 2.

Step-wise calculation:

1. For a quadratic ax² + bx + c, the sum and product of roots are `α + β = -b/a` and `αβ = c/a`.

2. Here a = 1, b = 1, c = –2, so α + β = –1 and αβ = –2.

3. Factor f(x): x² + x – 2 = (x + 2)(x – 1), so the zeros are –2 and 1.

4. Compute `1/α - 1/β = (β - α)/(αβ)`.

Using the found roots:

If α = –2 and β = 1,

`1/α - 1/β = (1/-2) - (1/1)`

= `-1/2 - 1`

= `-3/2`

If α = 1 and β = –2,

`1/α - 1/β = 1 - (-1/2)`

= `3/2`

The value is `±3/2` depending on which root is named α and which is named β. If α = –2 and β = 1, then `1/α - 1/β = -3/2`.