IIT JEE , AIEEE Forum - Mathematics , Algebra

AAKRITI

Find the values of parameter r so that x^{4_{- 4x}3 _{- 8x}2}_{^{+ r = 0 has (i) no real roots (ii) all roots real and distinct (iii) exactly 2 real roots}}

Sir kindly tell me such kind of questions are there in syllabus or not.

krishna.gopal

Hi aakriti I have already answered it in your other post but here i am pasting the reply again.

Before answering your problem i would like you ppl to look at roots of quadratic equation in a different way.

I want to understand the nature of roots of equation x^2 +(b/a)x+(c/a)=0

What i would do i will define f(x)=x^2 +(b/a)x+(c/a) and now i want to solve for f(x)=0.

You can appriciate that the form of f(x) will be such whose value is infinity at x= -infinity, will reach a minima in between and will again become infinite at x= +infinity.

Now if min(f(x) > 0 then there is no real solution for above equation

if min(f(x) = 0 then there is one real solution for above equation

if min(f(x) <0 then there are two real solution for above equation

Minima of f(x) occurs at x= -b/2a

and value of f(x) at this point is -b^2/(4a)+(c/a)

and hence i can find the nature of root. (The answer will be same if you do it by discriminant method.)

Now look at Aakriti's problem

f(x)=x^4-4x^3-8x^2+r=0

Putting f'(x)=4x^3-12x^2-16x=0

4x(x^2-3x-4)=0

This is zero at x=-1 (This will be first local minima as you move from -infinity to plus infinity.Call f(-1) as minima 1)

x=0 (This will be a local maxima. Call f(0) maxima)

and x=4 (This will be second local minima. Call f(4) minma 2)

If minima1>0 and minma2>0 Then function is always greater then zero and no real roots

If minma1<0, maxima>0 and minima2<0 tThen there are four distinct root

If one of the minima <0 and other >0 or if both minma and maxima are less than zer than only two real roots (Think of it by drawing graphs of such functions)

AAKRITI

Sir thank you very much for explaining me everything so clearly.

I am getting the following conditions:

4 real distinct roots- r

(0,3)

no real root- r > 128

only 2 real distinct roots- r

( 3, 128 )

only 3 real roots- r = 3

only 2 real equal roots- r = 128

atleast 2 real roots- r < 128

krishna.gopal

what about r<0

AAKRITI

For r < 0, we get exactly 2 real roots

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