Question

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

Answer 1

Given: Let α and β be the zeros of f(x) = x² + x − 2.

Step-wise calculation:

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

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

3. Compute `α/β + β/α = (α^2 + β^2)/(αβ)`.

4. α² + β² = (α + β)² – 2αβ 

= (–1)² − 2(–2) 

= 1 + 4

= 5

5. Therefore `α/β + β/α = 5/(-2)`

= `-5/2`