If f(x)=x^2+5bx+8c=0 has imaginary roots then which one is correct?
(a)5b+8c>-1
(b)5b+8c<-1
(c)5b+8c<-2
(d)5b+8c>-2.
 
(please also write the conditions necessary for roots of a quadratic equation to be imaginary)

Asked by rakshitreshu1997 | 31st Aug, 2015, 11:35: PM

Expert Answer:

f left parenthesis x right parenthesis equals x squared plus 5 b x plus 8 c G i v e n space t h a t space f left parenthesis x right parenthesis equals 0 space h a s space i m a g i n a r y space r o o t s. F o r space t h e space q u d r a t i c space e x p r e s s i o n comma space a equals 1 space left parenthesis w h i c h space i s space g r e a t e r space t h a n space z e r o right parenthesis. space H e n c e comma space t h e space g r a p h space o f space t h e space q u a d r a t i c space w i l l space o p e n space u p w a r d. A l s o comma space sin c e space t h e space q u a d r a t i c space e x p r e s s i o n space h a s space i m a g i n a r y space r o o t s comma space t h e space g r a p h space o f space t h e space q u a d r a t i c space e x p r e s s i o n space w i l l space n o t space i n t e r s e c t space X minus a x i s. I t space w i l l space a l w a y s space b e space a b o c e space X minus a x i s space f a c i n g space u p w a r d. H e n c e comma space f left parenthesis x right parenthesis space greater than space 0 comma space f o r space a l l space r e a l space v a l u e s space o f space apostrophe x apostrophe. H e n c e comma space f left parenthesis 1 right parenthesis space greater than space 0 rightwards double arrow 1 squared plus 5 b open parentheses 1 close parentheses plus 8 c greater than 0 rightwards double arrow 1 plus 5 b plus 8 c greater than 0 rightwards double arrow 5 b plus 8 c greater than negative 1 O p t i o n space left parenthesis a right parenthesis space i s space c o r r e c t. minus C o n d i t i o n space f o r space i m a g i n a r y space r o o t s colon D less than 0 rightwards double arrow open parentheses 5 b close parentheses squared minus 4 open parentheses 1 close parentheses open parentheses 8 c close parentheses less than 0 rightwards double arrow 25 b squared minus 32 c less than 0

Answered by satyajit samal | 1st Sep, 2015, 11:37: AM

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