If Equations #x^2+ax+b=0# & #x^2+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

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Answer 1 · 2017-04-25T11:26:25

see below

note:

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

that is

if #" "alpha" " & " "beta# are the roots of

#px^2+qx+r=0#

then

#alphabeta=-q/p#

#alphabeta =r/p#

#_______________________________________________#

#x^2+ax+b=0---(1)#

#x^2+cx+d=0---(2)#

let the common root be #alpha#

for eqn#(1)#

#alpha+alpha=-a#

#=>alpha=-a/2#

#" & "alpha^2=b#

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

then

#alpha+beta=-c#

#alphabeta=d#

#=>beta=d/alpha#

#:. alpha+d/alpha=-c#

#alpha^2+d=alpha(-c)#

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

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