If the equation x2 + abx + c = 0 & x2 + acx + v = 0, have a common root, prove that the other roots satisfy the equation x2 - a(b + c)x + a2bc = 0.

Let ∝ be the common roots of the given equations x2 + abx + c = 0 and x2 + 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 x2 + abx + c = 0 and  x2 + acx + b = 0 respectively.

∴ ∝β = c ⇒ β = ca  

and ∝γ = b ⇒ γ = ab

∴ βγ  = ca × ab = a2bc

β + γ = ca + ab = (a + bc)

The quadratic equation whose roots are β and γ is

x2 – (β + γ) x + βγ 

x2a(b + c)x + a2bc = 0

Hence, the other roots satisfy the equation x2a(b + c)x + a2bc = 0.

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