Question

if α and β are the zeros of ax² + bx + c, a ≠ 0 then verify  the relation between zeros and its cofficients

Answer 1

Since a and b are the zeros of polynomial ax² + bx + c.

Therefore, (x – α), (x – β) are the factors of the polynomial ax² + bx + c.

⇒ ax² + bx + c = k (x – α) (x – β)

⇒ ax² + bx + c = k {x² – (α + β) x + αβ}
⇒ ax² + bx + c = kx² – k (α + β) x + kαβ …(1)

Comparing the coefficients of x² , x and constant terms of (1) on both
sides, we get

a = k, b = – k (α + β) and c = kαβ

`⇒ α + β = \frac { -b }{ k } and αβ = \frac { c }{ k } `

`α + β = \frac { -b }{ a } and αβ = \frac { c }{ a } [∵ k = a]`

`"Sum of zeros" = (-b)/a="coefficient of x"/("coefficient of"x^2)`

`"Product of zeros"=c/d = "coefficient of x"/("coefficient of "x^2)`