Question

If α and β are the zeros of the quadratic polynomial f(x) = x² – 5x + 4, find the value of `1/α + 1/β - 2αβ`.

Answer 1

Given: If α and β are the zeros of f(x) = x² – 5x + 4, find `1/α + 1/β - 2αβ`.

Step-wise calculation:

1. For f(x) = x² – 5x + 4.

Compare with ax² + bx + c:

a = 1, b = –5, c = 4

Hence `α + β = -b/a = 5` and `αβ = c/a = 4`.

2. `1/α + 1/β = (α + β)/(αβ)`

= `5/4`

3. 2αβ = 2 × 4

= 8

4. Therefore `1/α + 1/β - 2αβ = 5/4 - 8`

= `5/4 - 32/4`

= `−27/4`

The value is `-27/4` (which is –6.75).