IIT JEE , AIEEE Forum - Mathematics , Algebra

magiclko

for how many integral values of a , the equation x² - (a+2)x + 2a-120, has integral roots??

Prabhuling

For integral values

b2 - 4ac >= 0

aditya_arora04

Hello my friend,

Do it as follows :

If the equation has integral roots, the b² - 4ac >= 0

Then, (a+2)² - 4( 2a-120 ) >= 0

a² + 4a + 4 - 8a + 480 >= 0

a² - 4a + 484 > 0 [ When you check Discriminant for

this equation, you will see that

it is < 0. So, = sign is removed ]

This equation is satisfied for all values of a.

So , ANS : a belongs to R

Please do tell me if i am wrong.

meghaguptaa

D has to b a perfect square also

magiclko

yeah D has also to be a perfect square, n a has exactly 32 values, as given in the answer, i dnt knw how they r getting it???

amar.gupta

The eq. Is:

x² - (a+2)x + (2a-120)=0

Let roots be A,B.

A=1/2[(a+2)+{(a+2)²-4*(2a-120)}^0.5]

B=1/2[(a+2)-{(a+2)²-4*(2a-120)}^0.5]

Let D=(a+2)²-4*(2a-120)

Or D=(a²+4+4*a-8*a+480)

=(a²-4*a+484)

Since A,B is an integer.

Hence D must be perfect square.

Let D=K² where K is a positive int.

Hence A=[a+2+K]/2 eq (1)

And B=[a+2-K]/2 eq (2)

Now D=K²

Or a²-4*a+484=K²

(a-2)²+480=K² eq (3)

or (K-a+2)(K+a-2)=480

Since K and a both are int.

The LHS will be multiple of two int.

480 can be written in multiples of two integers?.

(1,480) (2,240) (5,96) (3,160) (4,120) (10,48) (6,80) (15,32) (8,60) (20,24) (12,40) (30,16) ........

by comparing the two sides you can find the values of K and a.

for ex. (K-a+2)(K+a-2)=20*24

on comparing : K-a+2=20

K+a-2=24

On adding 2*K=44

Or K=22

From eq. (3) you will find that a=0 and 4

So for every value of K there will be two values of a.

Out of above 12 multiples, 4 does not give the intger valve of K.

So for rest 8 you can get 8 possible value of K and corresponding 16 values of a,

And so you can get the ans.

AAKRITI

yes sir , but you forgot to count corresponding negative values, so we get 32 values of a

amaron

i was able to come till (K-a+2)(K+a-2)=480
but was not able<found it too tiresome> to find number of values,
is there any easy way other than actual substitution to find number of values??
please reply.

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