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The-quadratic-equations-x-2-6x-a-0-and-x-2-cx-6-0-have-one-root-in-common-The-other-roots-of-the-first-and-second-equations-are-integers-in-the-ratio-4-3-Then-find-the-common-root-




Question Number 20118 by Tinkutara last updated on 22/Aug/17
The quadratic equations x^2  − 6x + a = 0  and x^2  − cx + 6 = 0 have one root in  common. The other roots of the first  and second equations are integers in  the ratio 4 : 3. Then, find the common  root.
Thequadraticequationsx26x+a=0andx2cx+6=0haveonerootincommon.Theotherrootsofthefirstandsecondequationsareintegersintheratio4:3.Then,findthecommonroot.
Answered by ajfour last updated on 22/Aug/17
Let the common root be α  αβ=a ,  α+β=6 , αγ=6 , α+γ=c  ⇒  β−γ = 6−c         (β/γ) = (a/6)=(4/3) ⇒   a=8  α+β=6  ; αβ=8 , ⇒   α=2,4  A check:  if α=2, then   β=4 , and γ=(3/4)β=3  if α=4, then β=2, and γ=(3/4)β=(3/2) ;  not an integer.  Hence α=2 .   c=α+γ =2+3=5 .
Letthecommonrootbeααβ=a,α+β=6,αγ=6,α+γ=cβγ=6cβγ=a6=43a=8α+β=6;αβ=8,α=2,4Acheck:ifα=2,thenβ=4,andγ=34β=3ifα=4,thenβ=2,andγ=34β=32;notaninteger.Henceα=2.c=α+γ=2+3=5.
Commented by Tinkutara last updated on 22/Aug/17
Thank you very much Sir!
ThankyouverymuchSir!

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