If the equation x^{2 }+ abx + c = 0 & x^{2 }+ acx + v = 0, have a common root, prove that the other roots satisfy the equation x^{2 }- a(b + c)x + a^{2}bc = 0.

Let ∝ be the common roots of the given equations *x*^{2} + *abx* + *c* = 0 and *x*^{2} + *acx* + *b* = 0.

∴ ∝^{2} + *ab*∝ + *c* = 0 ...(1)

∝^{2} + *ac*∝ + *b* = 0 ...(2)

Solving (1) and (2), we get

Let β and γ be the other roots of the equation *x*^{2} + *abx* + *c* = 0 and *x*^{2} + *acx* + *b* = 0 respectively.

∴ ∝β = *c* ⇒ β = *ca *

and ∝γ = *b* ⇒ γ = *ab*

∴ βγ = *ca* × *ab* = *a*^{2}*bc*

β + γ = *ca* + *ab* = (*a* + *bc*)

The quadratic equation whose roots are β and γ is

*x*^{2} – (β + γ) *x* + βγ

∴ *x*^{2} – *a*(*b* + *c*)*x* + *a*^{2}*bc* = 0

Hence, the other roots satisfy the equation *x*^{2} – *a*(*b* + *c*)*x* + *a*^{2}*bc* = 0.

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