If-the-roots-and-of-the-equation-ax-2-bx-c-0-are-real-and-of-opposite-sign-then-the-roots-of-the-equation-x-2-x-2-is-are-1-Positive-2-Negative-3-Real-and-opposite-si

Question Number 20052 by Tinkutara last updated on 21/Aug/17

If the roots α and β of the equation

{a x}^{2} + b x + c = 0 are real and of opposite

sign then the roots of the equation

α {(x - β)}^{2} + β {(x - α)}^{2} is / are

(1) Positive

(2) Negative

(3) Real and opposite sign

(4) Imaginary

Answered by ajfour last updated on 21/Aug/17

g i v e n α β < 0

s e c o n d e q u a t i o n r e f o r m e d i s

(α + β) x^{2} - 4 α β x + α β (α + β) = 0

D = 16 α^{2} β^{2} - 4 α β {(α + β)}^{2} > 0

f o r α β < 0

H e n c e r o o t s o f s e c o n d e q u a t i o n

γ, δ a r e r e a l .

γ δ = \frac{C}{A} = \frac{α β (α + β)}{α + β} = α β < 0

H e n c e γ a n d δ h a v e o p p o s i t e s i g n s .

(3) i s t h e c o r r e c t o p t i o n .

Commented by Tinkutara last updated on 21/Aug/17

Thank you very much Sir!