If Equations x^2+ax+b=0x2+ax+b=0 & x^2+cx+d=0x2+cx+d=0 have a common root and first equation have equal roots,solve Then prove that 2(b+d)=ac How to prove?solve

1 Answer
sjc
Apr 25, 2017

see below

Explanation:

note:

for this problem we will use the property of the sum and product of roots of a quadratic

that is

if " "alpha" " & " "beta α & β are the roots of

px^2+qx+r=0px2+qx+r=0

then

alphabeta=-q/pαβ=−qp

alphabeta =r/pαβ=rp

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x^2+ax+b=0---(1)x2+ax+b=0−−−(1)

x^2+cx+d=0---(2)x2+cx+d=0−−−(2)

let the common root be alphaα

for eqn(1)(1)

alpha+alpha=-aα+α=−a

=>alpha=-a/2⇒α=−a2

" & "alpha^2=b & α2=b

for the eqn(2)(2) let the second root be" "beta β

then

alpha+beta=-cα+β=−c

alphabeta=dαβ=d

=>beta=d/alpha⇒β=dα

:. alpha+d/alpha=-c

alpha^2+d=alpha(-c)

b+d=(-a/2)(-c)

:.2(b+d)=ac " as reqd."

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