Find the set that contains the real solution to an equation

[diredragon]

Homework Statement

$\frac{x+\sqrt3}{\sqrt{x}+\sqrt{x+\sqrt3}} + \frac{x-\sqrt3}{\sqrt{x}-\sqrt{x-\sqrt3}} = \sqrt{x}$
All real solutions to this equation are found in the set:
$a) [\sqrt3, 2\sqrt3), b) (2\sqrt3, 3\sqrt3), c) (3\sqrt3, 6), d) [6, 8)$

Homework Equations

3. The Attempt at a Solution

$\frac{x+\sqrt3}{\sqrt{x}+\sqrt{x+\sqrt3}} + \frac{x-\sqrt3}{\sqrt{x}-\sqrt{x-\sqrt3}} = \sqrt{x}$

$\frac{(x+\sqrt3)^2}{(\sqrt{x}+\sqrt{x+\sqrt3})^2} + \frac{(x-\sqrt3)^2}{(\sqrt{x}-\sqrt{x-\sqrt3})^2} + 2(\frac{x+\sqrt3}{\sqrt{x}+\sqrt{x+\sqrt3}}\frac{x-\sqrt3}{\sqrt{x}-\sqrt{x-\sqrt3}}) = x$

$\frac{(x+\sqrt3)^2(\sqrt{x}-\sqrt{x-\sqrt3})^2+(x-\sqrt3)^2(\sqrt{x}+\sqrt{x+\sqrt3})^2 + 2(x^2-3)((\sqrt{x}+\sqrt{x+\sqrt3})(\sqrt{x}-\sqrt{x-\sqrt3})}{(\sqrt{x}+\sqrt{x+\sqrt3})^2(\sqrt{x}-\sqrt{x-\sqrt3})^2}=x$

$\frac{(x+\sqrt{3})^2(\sqrt{x}-\sqrt{x-\sqrt{3}})^2+(x-\sqrt{3})^2(\sqrt{x}+\sqrt{x+\sqrt{3}})^2 + 2(x^2-3)((\sqrt{x}+\sqrt{x+\sqrt{3}})(\sqrt{x}-\sqrt{x-\sqrt{3}}))-(\sqrt{x}+\sqrt{x+\sqrt{3}})^2(\sqrt{x}-\sqrt{x-\sqrt{3}})^2x(\sqrt{x}+\sqrt{x+\sqrt{3}})^2(\sqrt{x}-\sqrt{x-\sqrt{3}})^2}=0$

$(x+\sqrt{3})^2(\sqrt{x}-\sqrt{x-\sqrt{3}})^2+(x-\sqrt{3})^2(\sqrt{x}+\sqrt{x+\sqrt{3}})^2 + 2(x^2-3)((\sqrt{x}+\sqrt{x+\sqrt{3}})(\sqrt{x}-\sqrt{x-\sqrt{3}}))-(\sqrt{x}+\sqrt{x+\sqrt{3}})^2(\sqrt{x}-\sqrt{x-\sqrt{3}})^2x=0$

there should be an easier way to do this rather than all i wrote + more and more. If you know, could you hint, rather than giving the answer?

 

[diredragon]

ok i have no idea how all this got messed up

 

[Samy_A]

First separately rationalizing the denominators in the two terms of the LHS, and only then squaring both sides, leeds to a manageable equation (that has to be squared again after simplifying).
There may be a smarter way to do this, though. For now it escapes me.

 

[Mark44]

ok i have no idea how all this got messed up

@diredragon, I fixed your LaTeX. Some obvious errors:
1. Don't use \Bigg or a variant -- That's not anything that our system recognizes, I don't believe. This could have been the source of all of those BBCode bold tags, but I'm not sure. In any case, you can't mix LaTeX and BBCode.
2. Don't write \sqrt3, as you had in many places -- this has to be \sqrt{3}.

Take a look at your first post to verify that my changes are what you intended.

 

[epenguin]

ok i have no idea how all this got messed up

And I have no idea what the question is.
OK, it's not stated but with a bit of goodwill...
It might have been solve that equation. In which case just a normal adding of fractions procedure already will give you some rationalization.
Or probably now I think about it the question could be to show that the pair of values all a, or the pair b, etc. are solutions. It would seem fairly easy to check any of these half dozen values. However I think it would probably be less work if you rationalized at least partly of the start anyway. Although you ought to be able to show that one of these pairs is a pair of solutions, I don't see how you can show that it is the only real solution pair unless you could get out of the equation the corresponding quadratic factor, and even then,... Well we will have to see what it looks like and if there is another factor maybe it's easy to show that has no real roots. But anyway get started. At the moment we don't even know what you've done.

 

[SammyS]

Homework Statement

$\frac{x+\sqrt3}{\sqrt{x}+\sqrt{x+\sqrt3}} + \frac{x-\sqrt3}{\sqrt{x}-\sqrt{x-\sqrt3}} = \sqrt{x}$
All real solutions to this equation are found in the set:
$a) [\sqrt3, 2\sqrt3), b) (2\sqrt3, 3\sqrt3), c) (3\sqrt3, 6), d) [6, 8)$

Homework Equations

3. The Attempt at a Solution

$\frac{x+\sqrt3}{\sqrt{x}+\sqrt{x+\sqrt3}} + \frac{x-\sqrt3}{\sqrt{x}-\sqrt{x-\sqrt3}} = \sqrt{x}$
...

there should be an easier way to do this rather than all i wrote + more and more. If you know, could you hint, rather than giving the answer?

The question only asks for the interval in which any solution (real) may be found.

I would start by finding the domain of each term.

 

[epenguin]

My unmathematicalness - those brackets are intervals - I thought they must be roots! I think the first steps will be the same though, and you will probably get something relatively simple.

 

[PeroK]

Here's an idea: let $x = \alpha y$ where $\alpha = \sqrt{3}$. Things simplify sufficiently without squaring. And you get a general result for $\alpha$ into the bargain.

 

[epenguin]

Now you have adjusted the formatting I can see your working. Yes that is unnecessarily complicated (Plus when you square things you have to be careful sometimes not to introduce false 'solutions'). A simpler way is what I previously suggested - I think I can see it works out to something relatively simple.

 

[Mark44]

My unmathematicalness - those brackets are intervals - I thought they must be roots!

The problem is not stated clearly, IMO.

All real solutions to this equation are found in the set:
$a) [\sqrt3, 2\sqrt3), b) (2\sqrt3, 3\sqrt3), c) (3\sqrt3, 6), d) [6, 8)$

A clearer question would be "All real solutions ... are found in the interval: "
Of course, an interval is a set, but their choices use interval notation.

 

[diredragon]

First separately rationalizing the denominators in the two terms of the LHS, and only then squaring both sides, leeds to a manageable equation (that has to be squared again after simplifying).
There may be a smarter way to do this, though. For now it escapes me.

After rationalising and squaring i get:
$3(x - \sqrt{3})^2(\sqrt{x} + \sqrt{x + \sqrt{3}})^2 + 3(x + \sqrt{3})^2(\sqrt{x} -\sqrt{x - \sqrt{3}})^2 - 6(x^2 - 3)(\sqrt{x} + \sqrt{x + \sqrt{3}})(\sqrt{x} - \sqrt{x -\sqrt{3}}) - 9x = 0$
I made out some assumption here:
$x \ge \sqrt{3}$ otherwise we don't get a real solutions because of $\sqrt{x - \sqrt{3}}$
But i can't get a uniform solution here. If this is the form of the equation after the simplification, and if indeed $x \ge \sqrt{3}$ than the $-9x$ never equals $0$. I am stuck

 

[PeroK]

After rationalising and squaring i get:
$3(x - \sqrt{3})^2(\sqrt{x} + \sqrt{x + \sqrt{3}})^2 + 3(x + \sqrt{3})^2(\sqrt{x} -\sqrt{x - \sqrt{3}})^2 - 6(x^2 - 3)(\sqrt{x} + \sqrt{x + \sqrt{3}})(\sqrt{x} - \sqrt{x -\sqrt{3}}) - 9x = 0$
I made out some assumption here:
$x \ge \sqrt{3}$ otherwise we don't get a real solutions because of $\sqrt{x - \sqrt{3}}$
But i can't get a uniform solution here. If this is the form of the equation after the simplification, and if indeed $x \ge \sqrt{3}$ than the $-9x$ never equals $0$. I am stuck

I recommend setting $x = \sqrt{3}y$ as in post #8. You should be able to get rid of all the $\sqrt{3}$ factors. Then it's a bit easier to work with the equation in $y$.

I must admit, I couldn't see the point in squaring things. The solution comes out without squaring.

 

[SammyS]

After rationalising and squaring i get:
$3(x - \sqrt{3})^2(\sqrt{x} + \sqrt{x + \sqrt{3}})^2 + 3(x + \sqrt{3})^2(\sqrt{x} -\sqrt{x - \sqrt{3}})^2 - 6(x^2 - 3)(\sqrt{x} + \sqrt{x + \sqrt{3}})(\sqrt{x} - \sqrt{x -\sqrt{3}}) - 9x = 0$
I made out some assumption here:
$x \ge \sqrt{3}$ otherwise we don't get a real solutions because of $\sqrt{x - \sqrt{3}}$
But i can't get a uniform solution here. If this is the form of the equation after the simplification, and if indeed $x \ge \sqrt{3}$ than the $-9x$ never equals $0$. I am stuck

What did you get after simplifying, but before squaring?

 

[epenguin]

I take back what I said before, there was something I didn't notice, and I don't think that combining the fractions as first step gives anything very simple.

You get some simplification bye rationalizing the fractions separately, but the result is still pretty ugly and intractable. I don't see any symmetry or any other feature of the expression that enables you to do it in any other way than by elaborating it as a polynomial, but this still looks complicated and nasty.

I invite the 0P to go back and check that he has transcribed the formula of problem exactly, and also I feel it is time we were sure what the question is! So please transcribe that exactly also.

 

[PeroK]

I take back what I said before, there was something I didn't notice, and I don't think that combining the fractions as first step gives anything very simple.

You get some simplification bye rationalizing the fractions separately, but the result is still pretty ugly and intractable. I don't see any symmetry or any other feature of the expression that enables you to do it in any other way than by elaborating it as a polynomial, but this still looks complicated and nasty.

I invite the 0P to go back and check that he has transcribed the formula of problem exactly, and also I feel it is time we were sure what the question is! So please transcribe that exactly also.

You should try $x = \sqrt{3}y$ and it simplifies to something more manageable!

 

[epenguin]

Tractable? I think I tracted it. But still requiring several squarings?

 

[PeroK]

Tractable? I think I tracted it. But still requiring several squarings?

No squaring! Just a tiny bit of algebraic sleight of hand.

 

[SammyS]

Remember: The problem statement merely asks for finding the interval in which the solution(s) is found.

 

[diredragon]

SammyS is right. I have written down the problem as i have read it and i agree that it asks for the interval in which the solution is found.
Im going to try the $x = \sqrt{3}y$ next

 

[epenguin]

Actually I'm not convinced that you need to simplify this expression at all to solve the problems. After all the arguments that you used to convince yourself that there are no real roots below √3 didn't really need this I think.

The algorithm guaranteed to tell you the number of real roots in an interval is that of Sturm or something like it. But that is pretty heavy calculation here. I think rather the question is inviting you to use your mathematical nous and a bit of luck to just solve it quicker way.

I'll give you a starter hint. It's quite easy to test if there is an odd number of routes in any interval.

That may then suggest things to try; at that point at least try to come up with a plan.

 

[ehild]

First separately rationalizing the denominators in the two terms of the LHS, and only then squaring both sides, leeds to a manageable equation (that has to be squared again after simplifying).

Did you get after rationalizing $$(x+\sqrt{3})^{3/2}+(x-\sqrt{3})^{3/2}=\sqrt{27x}$$?
After squaring once and simplifying, you get an upper limit for a real root.

 

[Samy_A]

Did you get after rationalizing $$(x+\sqrt{3})^{3/2}+(x-\sqrt{3})^{3/2}=\sqrt{27x}$$?
After squaring once and simplifying, you get an upper limit for a real root.

Yes, or using @PeroK 's suggestion and setting $x=\sqrt 3 y$,
$$(y+1)^{3/2}+(y-1)^{3/2}=3 \sqrt {y}$$
Following PeroK's second suggestion, I think one can deduce from this by sign analysis (of the difference of the expressions) in what interval solution(s) exist. I'm not sure that's what PeroK meant.
And for me, I just squared twice to find the root(s). Manageable, a little tedious, and above all very boring. I prefer PeroK's method.

 

[PeroK]

Yes, or using @PeroK 's suggestion and setting $x=\sqrt 3 y$,
$$(y+1)^{3/2}+(y-1)^{3/2}=3 \sqrt y$$

Following PeroK's second suggestion, I think one can deduce from this by sign analysis in what interval solution(s) exist. I'm not sure that's what PeroK meant.
And for me, I just squared twice to find the root(s). Manageable, a little tedious, and above all very boring. I prefer PeroK's method.

The initial substitution leads to:

$\frac{y+1}{\sqrt{y} + \sqrt{y+1}} + \frac{y-1}{\sqrt{y} - \sqrt{y-1}} = \sqrt{y}$

Then, multiplying through gives:

$y \sqrt{y} - \sqrt{y -1} - \sqrt{y+1} = -\sqrt{y} \sqrt{y+1} \sqrt{y-1}$

And then it's easy enough! The upper limit involves $\sqrt{5}$. I hope that's not giving too much away.

 

[SammyS]

The initial substitution leads to:

$\frac{y+1}{\sqrt{y} + \sqrt{y+1}} + \frac{y-1}{\sqrt{y} - \sqrt{y-1}} = \sqrt{y}$

Then, multiplying through gives:

$y \sqrt{y} - \sqrt{y -1} - \sqrt{y+1} = -\sqrt{y} \sqrt{y+1} \sqrt{y-1}$

And then it's easy enough! The upper limit involves $\sqrt{5}$. I hope that's not giving too much away.

I don't follow the algebra that takes your first equation to your second equation.

I get what Samy_A and ehild got. (They're equivalent.)

 

[PeroK]

I don't follow the algebra that takes your first equation to your second equation.

I get what Samy_A and ehild got. (They're equivalent.)

I think they are all equivalent. Quite a few terms cancel out on both sides.

 

[SammyS]

I think they are all equivalent. Quite a few terms cancel out on both sides.

I mean that the expressions Samy_A and ehild got are equivalent to each other.

 

[ehild]

And for me, I just squared twice to find the root(s). Manageable, a little tedious, and above all very boring. I prefer PeroK's method.

It is not that terrible. Lot of things cancel after both squarings.

 

[PeroK]

I mean that the expressions Samy_A and ehild got are equivalent to each other.

Yes, clearly. And they are both equivalent to my equation. The solution is the same in any case.

 

[Samy_A]

It is not that terrible. Lot of things cancel after both squarings.

Yes, it is certainly doable. That emboldened me to give the hint to try it this way.
But as the question is just about the interval(s), a shorter method is nicer.
Anyway, @diredragon now has received hints about 3 different methods (hope that doesn't confuse more than it helps).

 

[SammyS]

Yes, clearly. And they are both equivalent to my equation. The solution is the same in any case.

Never mind. I see it now.

 

[diredragon]

What did you get after simplifying, but before squaring?

After the initial setting i simplified to
$-(x+ \sqrt{3})(\sqrt{x} - \sqrt{x+ \sqrt{3}}) + (x- \sqrt{3})(\sqrt{x} + \sqrt{x- \sqrt{3}}) = \sqrt{3x}$

Did you get after rationalizing $$(x+\sqrt{3})^{3/2}+(x-\sqrt{3})^{3/2}=\sqrt{27x}$$?
After squaring once and simplifying, you get an upper limit for a real root.

I don't know how you got to this expression by rationalising.

The initial substitution leads to:

$\frac{y+1}{\sqrt{y} + \sqrt{y+1}} + \frac{y-1}{\sqrt{y} - \sqrt{y-1}} = \sqrt{y}$

Then, multiplying through gives:

$y \sqrt{y} - \sqrt{y -1} - \sqrt{y+1} = -\sqrt{y} \sqrt{y+1} \sqrt{y-1}$

And then it's easy enough! The upper limit involves $\sqrt{5}$. I hope that's not giving too much away.

I don't see how is the substitutes initial expression equivalent to this one. How do you know it is?

 

[Samy_A]

After the initial setting i simplified to
$-(x+ \sqrt{3})(\sqrt{x} - \sqrt{x+ \sqrt{3}}) + (x- \sqrt{3})(\sqrt{x} + \sqrt{x- \sqrt{3}}) = \sqrt{3x}$

I don't know how you got to this expression by rationalising.

In your expression, group the terms in $\sqrt{x+ \sqrt{3}}$ and $\sqrt{x- \sqrt{3}}$, and you will get the same expression as ehild and I got.
For example, you will get $(x+ \sqrt{3})( \sqrt{x+ \sqrt{3}})$, and that is nothing more than $(x+\sqrt{3})^{3/2}$.

I don't see how is the substitutes initial expression equivalent to this one. How do you know it is?

He multiplied and simplified. A lot of terms happen to cancel out.

 

[diredragon]

In your expression, group the terms in $\sqrt{x+ \sqrt{3}}$ and $\sqrt{x- \sqrt{3}}$, and you will get the same expression as ehild and I got.
For example, you will get $(x+ \sqrt{3})( \sqrt{x+ \sqrt{3}})$, and that is nothing more than $(x+\sqrt{3})^{3/2}$.He multiplied and simplified. A lot of terms happen to cancel out.

I see. I now get $(x- \sqrt{3})^3 + (x + \sqrt{3})^3 + 2(x^2 - 3)^{3/2} = 27x$
I tried the binomial expansion, doesn't get me anywhere, I am left with $2x^3$ term. How to solve the $0$ of this function?

 

[Samy_A]

I see. I now get $(x- \sqrt{3})^3 + (x + \sqrt{3})^3 + 2(x^2 - 3)^{3/2} = 27x$
I tried the binomial expansion, doesn't get me anywhere, I am left with $2x^3$ term. How to solve the $0$ of this function?

So you are trying it the "hard" way, by computing it.
Well, when you do the binomial expansions, you are left with the square root on one side and $9x-2x³$ on the other side. Square this again, do all the expansions, and you will be left with a rather easy equation for x.

 

[PeroK]

Alternatively:

$y \sqrt{y} - \sqrt{y -1} - \sqrt{y+1} = -\sqrt{y} \sqrt{y+1} \sqrt{y-1}$

$y \sqrt{y} - \sqrt{y -1} = \sqrt{y+1} (1 - \sqrt{y} \sqrt{y-1})$

Leads to an easier finish.