IIT JEE , AIEEE Forum - Mathematics , Algebra

chetan_kp

find all the possible values of ' a ' and ' b' for which if

is a root of the equation x² + ax + b = 0, then

²- 2 is also a root of the equation

johri_anshuman

read A=alpha
sum of roots = -a

=>A²-2 + A = -a ---- eq 1

product of roots = b
=>A(A²-2)=b ----- eq 2

the given equation is

x²+ax+b=0
put values of a and b

=>x²+x(2-A-A²)+A(A²-2)=0

as A is a root of this equation
=>A²+A(2-A-A²)+A(A²-2)=0

solve this equation to get 4 values of A and substitute the different values in equation (1) and (2) to get different values of a and b.

u'll get

a=2,b=0
a=0,b= -1
a= -4, b=4
a=2, b=1

nadeemoidu

@johri_anshuman
a=2,b=0 does not satisfy the conditions. Also you have missed some answers.

Here's my solution:

If A is a root of the equation , then A²-2 is also a root of the equation .

Note that if A ² -2 is a root of the equation of the equation then ( A ² -2 ) ² -2 should also be a root of the equation.

Case I)

( A ² -2 ) ² -2 = A

Note that A may or may not be equal to A ²-2 .

Solving we get A = 2 or A = -1 or A ² - A +1 =0

So both the roots are 2 or both the roots are -1 or the roots are 2 and -1
or the equation is x ² - x +1 =0

(a,b) can be (-4,4) , (-1,-2) , (2,1) ,( 1,-1)
Case II)

If A and B are the solutions,
B² -2 = A
and A² -2 = A

So a=0,b=-1 and a=0 , b=-4 are also answers.

Final answer is (-4,4) , (-1,-2) , (2,1) ,( 1,-1) ,(0,-1) ,(0,-4)

I forgot the second case during the olympiad.

me

according te me the answers r ----
(-4,4),(2,1),(1,-1) only
as accarging to question if @ is root then @^2 - 2 has to be other root
ie ----- if @^2-2 is root then (@^2-2)^2 - 2 also has to be the root.
and again we get {(@^2-2)^2 - 2} as root & so on.
but eq. can have at max 2 roots.
thus @^2 - 2=@ or (@^2-2)^2 - 2=@
which give above asnwers.
note here cobination of roots as 2 & -1 is not possible as if 2 is root then 4-2 should be other root thus eq. should have 2 as repeated root.

nadeemoidu

I dont understand the argument given above .

If 2 is a root of the equation it is not necessary that 4-2 should be THE OTHER root.
The question only says that 4-2 should also be a root of the equation , which is true for this case.

It is true for all the answers I have given above

nadeemoidu

Here are the complete solutions

http://www.isid.ac.in/~rbb/crmosol_07.pdf

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