Question

how can i easily find the roots of any quadratic equation...in α and β

Answer 1

ax² + bx +c  = 0

rewrite as x² + b/a  x + c/a  = 0    call  - b/a = b'    c/a  = c'
 x² - b' x + c' = 0  equaiton 1
IF α and β are the roots,
(x - α) (x - β) = 0    expand this expression then,  we get
x² - (α+ β) x +αβ = 0        comparew with the equation 1
α+β = b'  abd  αβ = product
Convert the equation in the form of equation 1  and then try to factor the c' so that the sum of the factors becomes minus of coefficient of x.

If some trial values fail, use the formula
Δ = √b² - 4ac    and    α, β  = (-b +- Δ)/2

Often trying simple values of x as x=1, -1, -2, -3, 2, 3 could give  a root directly.

Answer 2

any quadratic has roots from which we define sum of roots= alfa+beta
product = alfa x beta
alfa+ beta= -b/a   and alfa x beta= c/a   where   ax^2+bx+c is any quadratic.