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if α and β are the zeros of ax^2 + bx + c, a ≠ 0 then verify  the relation between zeros and its cofficients - Mathematics

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Sum

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

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Solution

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)`

Concept: Relationship Between Zeroes and Coefficients of a Polynomial
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