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?

Feb 21, 2017

$p = - 1$ and $m = \frac{1}{3}$

Explanation:

In a quadratic equation $a {x}^{2} + b x + c = 0$,

we have sum of roots as $- \frac{b}{a}$ and product of roots as $\frac{c}{a}$

Hence, as $p x \left(2 - x\right) = x - 4 m$

we have $- p {x}^{2} + 2 p x - x + 4 m = 0$

or $p {x}^{2} - \left(2 p - 1\right) x - 4 m = 0$

and as its roots are $\frac{a}{3}$ and $\frac{b}{3}$, we have

$\frac{a}{3} + \frac{b}{3} = \frac{2 p - 1}{p}$ or $\frac{2 p - 1}{p} = \frac{a + b}{3} = \frac{9}{3} = 3$

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

Further $\frac{a}{3} \times \frac{b}{3} = \frac{- 4 m}{p} = 4 m$ (as $p = - 1$)

Hence $4 m = \frac{a b}{9} = \frac{12}{9} = \frac{4}{3}$ i.e. $m = \frac{1}{3}$