IIT JEE , AIEEE Forum - Mathematics , Algebra

subhashini90

Q.1 If

and

are the roots of the equation x²+ 2ax + b = 0,construct a quad. equation with rational coefficient and one of whose roots is

+

²+,given that

² +

² is not a perfect square.

Q.2 If

and

are roots of ax² + bx + c = 0,find the value of (a

+b)^-3 + (a

+ b)^-3

Answers for both questions are as follows :
Q.1 x² + 4ax + 2b = 0
Q.2 (b³ - 3abc)/a³ c³

Thanx!!

Moderator

Dear Subhashini,

Experts only answer one question at a time, please let us know which question needs to be answered first and post other queries again.

~ moderator

subhashini90

Thanx a lot fer tellin...i didnt know..!!

well..heres my question

If

and

are the roots of the equation x²+ 2ax + b = 0,construct a quad. equation with rational coefficient and one of whose roots is

+

²+,given that

² +

² is not a perfect square.

yahiyafirdous

Q1 INCOMPLETE

Q2

+= -b/a

=c/a

(a+b)(a +b)= b² +ab(+) + a²

=b² - b² +ac=ac

² + ²=(+)² - 2= (b/a)²+(c/a)²

Now given expression

= [a +b+a + b][a²² +a²² - (a+b)(a +b)]/(a+b)³(a +b)³

=(- b + 2b)(b² -2ac-ac)/(ac)³=(b³-3abc)(ac)³ Ans

subhashini90

If

and

are the roots of the equation x²+ 2ax + b = 0,construct a quad. equation with rational coefficient and one of whose roots is

+

²+

²
given that

² +

² is not a perfect square.

DON007

²hey .............look...............

using above equation we get...........

+

= -2a........(1)

=b.....(2)

now we have to quad. equation with rational coefficient and one of whose roots is

+

²+,given that

² +

² is not a perfect square.

so..........we have studied if one root is a+sqrtb

then other root is a-sqrtb

so other root of this equation is.............

+

-

²+².

so forming the equation ........

x² - (sum of roots)x + product of roots=0

x²- [(

+

²+² ) + (

+

-

²+²)]x + [ (+

+

²+ ² ) (

+

²+2 ) ]=0

_{x²- 2 (+)x +( +)}2 _{- (}2₊2₎₌₀

_x2 _{- 2(-2a)x + 2}₌₀

_x2_{+4ax + 2b=0 ............ans}

_thanks

iitkgp_bipin

Good job done DON.

The roots are (

+

²+² ) and (

+

-

²+²)

Sum of roots = 2(

+

) = -4a
Product of roots = (

+

)² - (

² +

²) = 2

= 2b

Hence the quadratic equation would be

x² + 4ax + 2b = 0

faith

but isnt it true only for eqns with rational coefficients

" if one root is a+sqrtb
then other root is a-sqrtb
"

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