If one root of the quadratic equation #ax^2+bx+c=0# is equal to #n^(th)# power of the other, then show that : #(ac^n)^(1/(n+1))+(a^n c)^(1/(n+1)) + b=0#?

1 Answer
P dilip_k
Jul 20, 2017

If one root of the quadratic equation #ax^2+bx+c=0# is equal to #n^(th)# power of the other, then show that :
#(ac^n)^(1/(n+1))+(a^n c)^(1/(n+1)) + b=0#?

Let one root be #alpha# then other will be #alpha^n#

So we can write

#alpha^n +alpha=-b/a#

and

#alpha^(n+1)=c/a#

#=>alpha=(c/a)^{1/(n+1))#

Now one root of the quadratic equation #ax^2+bx+c=0# being #alpha#. we can write

#aalpha^2+balpha+c=0#

#=>aalpha+b+c/alpha=0#

#=>c/alpha+aalpha+b=0#

#=>c/(c/a)^{1/(n+1))+a(c/a)^{1/(n+1))+b=0#

#=>c(a/c)^{1/(n+1))+a(c/a)^{1/(n+1))+b=0#

#=>((c^(n+1)a)/c)^{1/(n+1))+((a^(n+1)c)/a)^{1/(n+1))+b=0#

#=>(ac^n)^{1/(n+1))+(a^nc)^{1/(n+1))+b=0#

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