Question

If a quadratic polynomial y=ax^{2}+bx +c intersects x axis at α and β, then

Answer 1

Answer:

αβ = c/a

α+β = -b/a

Step-by-step explanation:

If a quadratic polynomial y=ax^{2}+bx +c intersects x axis at α and β, then

Polynomial intersect x axis at α & β

x axis intersection mean y = 0

=> ax² + bx + c = c  for α & β

=> aα² + bα + c = 0    - Eq 1

=> aβ² + bβ + c = 0   - Eq 2

β×Eq1  - α×Eq 2

=> aα²β + bαβ + cβ - aαβ² - bαβ - cα = 0

cancelling bαβ

=>  aα²β + cβ - aαβ² - cα = 0

=> aαβ(α-β) - c(α-β) = 0

=> (α-β)(aαβ - c) = 0

=> αβ = c/a  

Eq 1 - Eq2

aα² + bα - aβ² - bβ = 0

=> a(α²-β²) + b(α-β) = 0

=> a(α+β)(α-β) + b(α-β) = 0

=> (α-β)( a(α+β) + b) = 0

=> α+β = -b/a