Given-that-2x-2-3px-2q-and-x-2-q-have-a-common-factor-x-a-where-p-q-and-a-are-none-zero-constants-show-that-9p-2-16q-0-

Question Number 4707 by 314159 last updated on 28/Feb/16

G i v e n t h a t 2 x^{2} + 3 p x - 2 q a n d x^{2} + q h a v e a

c o m m o n f a c t o r x - a, w h e r e p, q a n d a a r e n o n e

z e r o c o n s t a n t s, s h o w t h a t 9 p^{2} + 16 q = 0 .

Commented by prakash jain last updated on 26/Feb/16

If equation a_{1} x^{2} + b_{1} x + c_{1} = 0

and a_{2} x^{2} + b_{2} x + c_{2} = 0 have one common

root α then

\frac{α^{2}}{b_{1} c_{2} - b_{2} c_{1}} = \frac{α}{a_{2} c_{1} - a_{1} c_{2}} = \frac{1}{a_{1} b_{2} - a_{2} b_{1}}

f o r t h e g i v e n q u e s t i o n α = a

a_{1} = 2, b_{1} = 3 p, c_{1} = - 2 p

a_{2} = 1, b_{2} = 0, c_{2} = q

a = \frac{- 2 p - 2 q}{- 3 p} = \frac{2 (p + q)}{3 p}

a^{2} = \frac{3 p q}{- 3 p} = - q

Commented by prakash jain last updated on 26/Feb/16

I think the first equation should be

2 x^{2} + 3 p x - 2 q = 0 to get the required result .

t h e n

a = \frac{4 q}{3 p}

a^{2} = - q

16 q^{2} = - 9 p^{2} q

9 p^{2} + 16 q = 0