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If-the-equations-x-2-ax-b-0-and-x-2-bx-a-0-have-a-common-root-then-the-numerical-value-of-a-b-is-

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Question Number 87275 by unknown last updated on 03/Apr/20
If the equations x^2 +ax+b=0 and x^2 +bx+a=0 have a common root, then the numerical value of a+b is
$$\mathrm{If}\:\:\mathrm{the}\:\mathrm{equations}\:{x}^{\mathrm{2}} +{ax}+{b}=\mathrm{0}\:\mathrm{and}\: \\ $$$${x}^{\mathrm{2}} +{bx}+{a}=\mathrm{0}\:\mathrm{have}\:\mathrm{a}\:\mathrm{common}\:\mathrm{root}, \\ $$$$\mathrm{then}\:\mathrm{the}\:\mathrm{numerical}\:\mathrm{value}\:\mathrm{of}\:{a}+{b}\:\mathrm{is} \\ $$
Answered by Rio Michael last updated on 03/Apr/20
let the common root be α ⇒ α^2 + αa + b = 0 also α^2 + αb + a = 0 ⇒ when x = α α^2 + αa + b = α^2 + αb + a αa + b = αb + a αa − αb = a − b α (a−b) = (a−b) ⇒ α = 1 from x^2 + ax + b = 0 at x = 1 1 + a + b = 0 ⇒ a + b = −1 please check sir
$$\mathrm{let}\:\mathrm{the}\:\mathrm{common}\:\mathrm{root}\:\mathrm{be}\:\alpha \\ $$$$\:\Rightarrow\:\:\alpha^{\mathrm{2}} \:+\:\alpha{a}\:+\:{b}\:=\:\mathrm{0} \\ $$$$\:\mathrm{also}\:\:\alpha^{\mathrm{2}} \:+\:\alpha{b}\:+\:{a}\:=\:\mathrm{0} \\ $$$$\Rightarrow\:\mathrm{when}\:{x}\:=\:\alpha \\ $$$$\:\:\:\alpha^{\mathrm{2}} \:+\:\alpha{a}\:+\:{b}\:=\:\alpha^{\mathrm{2}} \:+\:\alpha{b}\:+\:{a} \\ $$$$\:\:\:\:\:\:\:\:\:\:\alpha{a}\:+\:{b}\:=\:\alpha{b}\:+\:{a} \\ $$$$\:\:\:\:\:\:\:\:\:\alpha{a}\:−\:\alpha{b}\:=\:{a}\:−\:{b} \\ $$$$\:\:\:\:\:\:\:\:\alpha\:\left({a}−{b}\right)\:=\:\left({a}−{b}\right)\:\Rightarrow\:\alpha\:=\:\mathrm{1} \\ $$$$\:\:\:\:\:\mathrm{from}\:{x}^{\mathrm{2}} \:+\:{ax}\:+\:{b}\:=\:\mathrm{0}\:\mathrm{at}\:{x}\:=\:\mathrm{1} \\ $$$$\:\:\:\:\:\:\mathrm{1}\:+\:{a}\:+\:{b}\:=\:\mathrm{0}\:\Rightarrow\:{a}\:+\:{b}\:=\:−\mathrm{1} \\ $$$$\boldsymbol{\mathrm{please}}\:\boldsymbol{\mathrm{check}}\:\boldsymbol{\mathrm{sir}} \\ $$$$ \\ $$
Commented by mr W last updated on 03/Apr/20
please don′t answer a question which is red marked! it is red marked because it should be removed.
$${please}\:{don}'{t}\:{answer}\:{a}\:{question}\:{which} \\ $$$${is}\:{red}\:{marked}!\:{it}\:{is}\:{red}\:{marked}\:{because} \\ $$$${it}\:{should}\:{be}\:{removed}. \\ $$
Commented by Rio Michael last updated on 03/Apr/20
really sir? sorry i didn′t know, is there some group rules for that?
$$\mathrm{really}\:\mathrm{sir}?\:\mathrm{sorry}\:\mathrm{i}\:\mathrm{didn}'\mathrm{t}\:\mathrm{know},\:\mathrm{is}\:\mathrm{there}\:\mathrm{some}\:\mathrm{group} \\ $$$$\mathrm{rules}\:\mathrm{for}\:\mathrm{that}? \\ $$
Commented by mr W last updated on 03/Apr/20
A post is red marked through the menu “Flag post as inappropriate”. usually because it is nonsense, repeated. there are no special rules in this forum. that means we should keep the forum clean, by each of us.
$${A}\:{post}\:{is}\:{red}\:{marked}\:{through}\:{the} \\ $$$${menu}\:“{Flag}\:{post}\:{as}\:{inappropriate}''. \\ $$$${usually}\:{because}\:{it}\:{is}\:{nonsense},\: \\ $$$${repeated}.\:{there}\:{are}\:{no}\:{special}\:{rules}\: \\ $$$${in}\:{this}\:{forum}.\:{that}\:{means}\:{we}\:{should} \\ $$$${keep}\:{the}\:{forum}\:{clean},\:{by}\:{each}\:{of}\:{us}. \\ $$
Commented by Rio Michael last updated on 04/Apr/20
you are right sir, and am greatful to you
$$\mathrm{you}\:\mathrm{are}\:\mathrm{right}\:\mathrm{sir},\:\mathrm{and}\:\mathrm{am}\:\mathrm{greatful}\:\mathrm{to}\:\mathrm{you} \\ $$

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