62t=3x-3
t/x = 3-3/x
x1/3 + (t-x)1/3 = (4t)1/3
1+ (t/x-1)1/3 = (4t/x)1/3
taking cube
1+ 3(t/x-1)1/3 + 3(t/x-1)2/3 + (t/x-1) = 4t/x
3(t/x-1)1/3 + 3(t/x-1)2/3 = 3t/x
(t/x-1)1/3 + (t/x-1)2/3 = t/x
(t/x-1) + (t/x-1)2 +3(t/x-1)(t/x) = (t/x)3
(t/x-1) + (t/x-1)2 +3(t/x-1)(t/x) = (t/x)3
(t/x-1)t/x +3(t/x-1)(t/x) = (t/x)3
4(t/x-1)(t/x) = (t/x)3
Thus , t/x=0 or
4(t/x-1) = (t/x)2
(t/x)2 -4t/x +4 =0
t/x=2
thus solved :)
62t/x = 3-3/x
3-3/x= 0
x=1
3-3/x =2
3/x=1
x=3
so x=1, 3 are the solutions!
24Thanks for reply..
Just one question.:: Can these type of questions come in JEE ?? I mean to say that should I practice more of such type or not. ??
62frankly i think i have messed up the solution.. so it is looking dirty and tough..
JEee will have more simpler problems (or which are solved in lesser no of steps.!)
62x1/3 + (2x-3)1/3 = (12x-12)1/3
take cube
x+2x-3 + {x(2x-3)}1/3{x1/3 + (2x-3)1/3} = 12x-12
thus,
x+2x-3 + {x(2x-3)}1/3 (12x-12)1/3= 12x-12
thus,
{x(2x-3)}1/3 (12x-12)1/3 = 9x-9
Now this is simple? (oops simpler?)
To try to make the solution simple, i made it more difficult :D
341Let a = x1/3 and b = (2x-3)1/3
Then the equation is
a1/3+b1/3 - 41/3(a+b)1/3 = 0
We know that if p+q+r =0, then p3+q3+r3 = 3pqr
Hence a+b-4(a+b) = -3.(4ab)1/3(a+b)1/3 or
(a+b) = (4ab)1/3(a+b)1/3
Now either a+b = 0 i.e. x = 1 or
(a+b)2/3 = (4ab)1/3
Cubing both sides, we get
(a+b)2 = 4ab or (a-b)2 = 0 or a = b
Hence x = 2x-3 giving x = 3 as the other solution
62great man ....
I was trying to simplify it.. and made a big mess of it :D
Godo that you simplified it :)
341There are several of these kind of problems. One that comes to mind immediately is:
Solve (6-x)4+(x-8)4 = 16
Unless you spot that this method can be used, it will be a very messy problem.
I will see if I can hunt down other problems of this variety
341and since this is eureka's thread I want to just say that see we will be eager to help you out bcos u r a bro (applies to sis also) who is on the same path. But when we put effort and then no one says a thing it will be like wasted. Now Nishant brother said something so it makes me look if i can add something more. You get more cooperation just by being nice