Roots of the equation $px(2-x)=x-4m$ are $a/3$ an $b/3$ and while $a+b=9$ , $ab=12$ . Find $p$ and $m$ ?

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Answer 1 · 2017-02-21T09:48:29

$p=-1$ and $m=1/3$

In a quadratic equation $ax^2+bx+c=0$ ,

we have sum of roots as $-b/a$ and product of roots as $c/a$

Hence, as $px(2-x)=x-4m$

we have $-px^2+2px-x+4m=0$

or $px^2-(2p-1)x-4m=0$

and as its roots are $a/3$ and $b/3$ , we have

$a/3+b/3=(2p-1)/p$ or $(2p-1)/p=(a+b)/3=9/3=3$

Hence $2p-1=3p$ or $p=-1$

Further $a/3xxb/3=(-4m)/p=4m$ (as $p=-1$ )

Hence $4m=(ab)/9=12/9=4/3$ i.e. $m=1/3$

Roots of the equation px(2-x)=x-4mpx(2-x)=x-4m are a/3a/3 an b/3b/3 and while a+b=9a+b=9, ab=12ab=12. Find pp and mm?