Find the set of all possible real values of a such that the linear equality (x-(a-1))(x-(a squared+ 2) )<0 holds for all x that belongs to(-1,3).
 
Kindly give a step wise explanation.

Asked by kumar.ashlesha | 14th Jun, 2016, 01:13: PM

Expert Answer:

begin mathsize 12px style open parentheses straight x minus open parentheses straight a minus 1 close parentheses close parentheses open parentheses straight x minus open parentheses straight a squared plus 2 close parentheses close parentheses less than 0

Here comma space we space will space use space wavy minus curve space method comma space to space locate space the space roots space of space
open parentheses straight x minus open parentheses straight a minus 1 close parentheses close parentheses open parentheses straight x minus open parentheses straight a squared plus 2 close parentheses close parentheses equals 0 space on space straight a space number space line.
The space roots space of space open parentheses straight x minus open parentheses straight a minus 1 close parentheses close parentheses open parentheses straight x minus open parentheses straight a squared plus 2 close parentheses close parentheses equals 0 space are space left parenthesis straight a minus 1 right parenthesis space and space left parenthesis straight a squared plus 2 right parenthesis
To space know space which space of space straight a squared plus 2 space and space left parenthesis straight a minus 1 right parenthesis space is space larger comma
we space take space straight a squared plus 2 minus open parentheses straight a minus 1 close parentheses
equals straight a squared minus straight a plus 3
This space is space straight a space quadratic space in space apostrophe straight a apostrophe comma space with space straight D less than 0 space and space coefficient space of space straight a squared space positive.
So space straight a squared minus straight a plus 3 space is space always space positive
straight a squared minus straight a plus 3 greater than 0
rightwards double arrow straight a squared plus 2 minus open parentheses straight a minus 1 close parentheses greater than 0
rightwards double arrow straight a squared plus 2 greater than straight a minus 1
So space on space straight a space number space line comma space straight a squared plus 2 space will space be space to space the space right space of space straight a minus 1.

end style
S o space f o r space open parentheses x minus open parentheses a minus 1 close parentheses close parentheses open parentheses x minus open parentheses a squared plus 2 close parentheses close parentheses less than 0
W e space g e t
x element of open parentheses a minus 1 comma space a squared plus 2 close parentheses
B u t comma space i t apostrophe s space g i v e n space t h a t space apostrophe x apostrophe space b e l o n g s space t o space left parenthesis negative 1 comma space 3 right parenthesis.
S o space a minus 1 space a n d space a squared plus 2 space s h o u l d space b e space s u c h space t h a t space left parenthesis negative 1 comma space 3 right parenthesis space i s space e n t i r e l y space i n s i d e space t h e m
o n space t h e space n u m b e r space l i n e space a s space s h o w n space b e l o w
begin mathsize 12px style So space straight a minus 1 less than negative 1 space and space straight a squared plus 2 greater than 3
rightwards double arrow straight a less than 0 space and space straight a squared greater than 1
rightwards double arrow straight a less than 0 space and space straight a less than negative 1 space or space straight a greater than 1
So space the space values space of space apostrophe straight a apostrophe space satisfying space both space the space conditions space are space straight a less than negative 1

Hence comma space for space all space values space of space straight a less than negative 1 comma space the space given space inequality space will space be space satisfied
for space all space values space of space apostrophe straight x apostrophe space in space the space interval space left parenthesis negative 1 comma space 3 right parenthesis. end style

Answered by satyajit samal | 15th Jun, 2016, 03:20: PM