GMAT Problem Solving Discussions

[jha16june]

Originally Posted by sillyfool

hi rajkumar
i guess the answer to the question is e
when you take square of a no. and then take square root it can have either of the two values i mean +x or -x so it becomes |x| as x <>0 so divided by x becomes the option e

hope this helps
Sillyfool

Originally Posted by Bhavin;1112995

yes...the ans is E...but statement in bold is incorrect...bcoz square of a no is always +ve and square root of it is only positive..i.e 25^1/2 is only +5 and not -5...

lets plug in..if x=-5. x^2=25 & 25^1/2=5
i.e [(x^2)^1/2

/x is 5/-5 = -1 i.e |x|/x

if x=5, x^2=25 & 25^1/2=5
i.e [(x^2)^1/2]/x is 5/5=1 i.e |x|/x

Hence |x|/x satisfies for +ve n -ve nos...Ans E

Well Bhavin is absolutely correct! These are basics. sqrt(25) is only and only +5 and nver... -5. In the question we are taking |x| to ensure that it stays +ve, irrespective of what x we choose.
When u r asked what is the value of x, given that x^2 = 25, then x=+5, and -5 are correct. There is no convention and all. And not to forget that sqrt of a negative number is not defined at all.

[sillyfool]

Originally Posted by jha16june

Well Bhavin is absolutely correct! These are basics. sqrt(25) is only and only +5 and nver... -5. In the question we are taking |x| to ensure that it stays +ve, irrespective of what x we choose.
When u r asked what is the value of x, given that x^2 = 25, then x=+5, and -5 are correct. There is no convention and all. And not to forget that sqrt of a negative number is not defined at all.

i disagree to your statement in bold
i know these are basics
but bro what do you mean when you say that square root of 25 is only +5
i compeltely disagree. what will happen to bechaaara -5 is bechaare ka kya kasoor hai

when you take square of -5 you write it as 25 but you say when you will take square root
you will forget about -5 . you are doing wrong with poor bechaara -5
Please consider again. I guess -5 also has the right to be called the square root of 25

[jha16june]

Originally Posted by sillyfool

i disagree to your atatement in bold
i know these are basics
but bro what do you mean when you say that square root of 25 is only +5
i compeltely disagree. what will happen to bechaaara -5 is bechaare ka kya kasoor hai

when you take square of -5 you write it as 25 but you say when you will take square root
you will forget about -5 . you are doing wrong with poor bechaara -5
Please consider again. I guess -5 also has the right to be called the square root of 25

Well u r mistaking in formulating the question. finding sqrt(25) is not same as finding all those x for which x^2 = 25.
Well I cannot reply any more to this discussion. If u don agree, u have every right to disagree. Probably u surf a bit and make ur own decision.

the definition goes like this:

sqrt(x^2) = x if x>=0 and , = -x, if x<0. so sqrt(x^2) = |x|.

[sillyfool]

Originally Posted by jha16june

Well u r mistaking in formulating the question. finding sqrt(25) is not same as finding all those x for which x^2 = 25.
Well I cannot reply any more to this discussion. If u don agree, u have every right to disagree. Probably u surf a bit and make ur own decision.

well i again disagree with your statement in bold

can you tell me the diff in the statement
i mean you have written 2 ways of doing the same finding square root of 25
what is the diff?
can you please explain?
if you dont wanna post you can PM me

Sillyfool

[bhavin422]

Originally Posted by jha16june

Well Bhavin is absolutely correct! These are basics. sqrt(25) is only and only +5 and nver... -5. In the question we are taking |x| to ensure that it stays +ve, irrespective of what x we choose.
When u r asked what is the value of x, given that x^2 = 25, then x=+5, and -5 are correct. There is no convention and all. And not to forget that sqrt of a negative number is not defined at all.

Originally Posted by sillyfool

i disagree to your atatement in bold
i know these are basics
but bro what do you mean when you say that square root of 25 is only +5
i compeltely disagree. what will happen to bechaaara -5 is bechaare ka kya kasoor hai

when you take square of -5 you write it as 25 but you say when you will take square root
you will forget about -5 . you are doing wrong with poor bechaara -5
Please consider again. I guess -5 also has the right to be called the square root of 25

hey bro...jha is correct....sqrt of non negative integer is never negative..

u can also look at this way...
A implies B does not mean B implies A.
i.e square of -5 implies 25....does not mean root of 25 implies -5...

[abchekstylo]

hi sillyfool,
like i already posted long back in this thread, we do have to consider the positive as well as the negative root of a positive integer.
In fact that's a common GMAT trap.There were some questions that checked just this fact and as u can guess, even those books(good ones like Kaplan 800) highlighted the fact that there are 2 roots and we must consider them.

So 25 has 2 roots:+5 and -5(like u already know)
You can take my word for it.

Cheers!

[eric.segal1]

ok sillyfool let me try to clear it
if the Qn says x^2=25
then x= +/- srt25
which implies x=+/-5
but if the Qn is x=srt25
then x=5
you must understand here that the +/- comes when you shift a square across the equality and is in no way associated with Square root as such.
this is a mathematical convention just as 0!=1
u cannot deny this convention. Its a mathematical law dude.
hope it helps..........

[padfoot]

and should i jump in too
let x=square root of 25
so that x2-25=0 ....(1)
taking f(x)=x2-25 , newton raphson method gives
xn+1 = xn - [ f(xn) /f '(xn) ]
=xn - [ x2n - 25 /2xn ]
= ½ [ xn + (25 / xn ) ] .....(2)
now since f(4) = -9 , f(6) = 11 ,
and also since f(-4) = -9 and f(-6) = 11
a root of (1) lies between 4 and 6 and -4 and -6
Case 1
therefore , taking x0 = 4.5 , equ (2) gives
x1= ½ [ x0 + (25 /x0 ) ]
= ½ [ 4.5 +(25/4.5)]
= 5.0277778
x2= ½ [ x1 + (25 /x1 ) ]
= ½ [5.0277778 +(25/5.0277778 )]
= 5.0000767
x3= ½ [ x2 + (25 /x2 ) ]
= ½ [5.0000767 +(25/5.0000767 )]
= 5
x4=½ [ x3 + (25 /x3 ) ]
= ½ [5 +(25/5 )]
= 5
since x3=x4
we take square root of 25 = 5
Case 2
therefore , taking x0 = -4.5 , equ (2) gives
x1= ½ [ x0 + (25 /x0 ) ]
= ½ [ -4.5 +(25/-4.5)]
= -5.0277778
x2= ½ [ x1 + (25 /x1 ) ]
= ½ [-5.0277778 +(25/-5.0277778 )]
= -5.0000767
x3= ½ [ x2 + (25 /x2 ) ]
= ½ [5.0000767 +(25/5.0000767 )]
= -5
x4=½ [ x3 + (25 /x3 ) ]
= ½ [-5 +(25/-5 )]
= -5
since x3=x4
we take square root of 25 = -5

Hence there are 2 cases -5 and 5 . Thus, every positive number has two square roots, one positive and the other negative.Hope Helps !! Silly,ab,eric lets stop the love story here, now , in GMAT its important to refer to both the roots. So this is very clear .

[padfoot]

and further to end this discussion which might be useful for you all find this problem

solve

x^2 - 5x + 2*(SQRT(x^2 - 5x +3 )) = 12

[abchekstylo]

Originally Posted by padfoot

and further to end this discussion which might be useful for you all find this problem

solve

x^2 - 5x + 2*(SQRT(x^2 - 5x +3 )) = 12

x^2 - 5x + 2*(√ (x^2 - 5x +3 )) = 12 ...................(1)
Let (x^2 - 5x +3) = A^2 ...................(2)
√(A^2) = |A|
So (x^2 - 5x) = A^2 - 3



So the equation 1 becomes
A^2 - 3 + 2|A| = 12
A^2 + 2|A| – 15 = 0
|A|^2+ 2|A| – 15 = 0 since A^2=|A|^2
(|A|+5)(|A|-3)=0
|A|=-5 or 3 ...................(3)


Now, |A| cannot be -5 according to definition of modulus

So |A| = 3
So |A|^2 = 9
A^2 = 9

From (2) we get
x^2 - 5x +3 = 9 ...................(4)
x^2 - 5x -6 = 0
(x-6)(x+1)=0
x= 6 , -1