B Given an equation, find the value of this expression

  • B
  • Thread starter Thread starter littlemathquark
  • Start date Start date
  • Tags Tags
    Algebra math
AI Thread Summary
The discussion revolves around solving the equation -1 + (2/(1-x)) = √3 and finding the value of (2x/(1-x²))√3. Multiple methods are proposed, including direct substitution for x and using trigonometric identities with x = tan(α). Participants express confusion over the claim that the expression evaluates to 1, as their calculations yield different results. Several solutions are explored, including geometric interpretations and algebraic manipulations, ultimately leading to the conclusion that the expression can be evaluated to 1 through various methods. The conversation highlights the complexity of the problem and the need for clarity in the steps taken to reach the solution.
littlemathquark
Messages
204
Reaction score
26
TL;DR Summary
İf ##-1+\dfrac{2}{1-x}=\sqrt 3## then find ##\dfrac{2x}{1-x^2}\cdot \sqrt 3##
İf ##-1+\dfrac{2}{1-x}=\sqrt 3## then find ##\dfrac{2x}{1-x^2}\cdot \sqrt 3##

I can solve it with two ways:
1) directly by finding ##x##
2) Let ##x=\tan\alpha## and go on and answer is ##1.##

But I need another way. Are there an another way?
 
Mathematics news on Phys.org
littlemathquark said:
I can solve it with two ways:
1) directly by finding ##x##
2) Let ##x=\tan\alpha## and go on and answer is ##1.##

But I need another way. Are there an another way?
How are you getting an "answer" of 1? This can't be the value of x as it would make the equation undefined. The value I get for x doesn't give a value of 1 for the expression ##\frac{2x}{1 - x^2}\sqrt 3##.
 
Mark44 said:
How are you getting an "answer" of 1? This can't be the value of x as it would make the equation undefined. The value I get for x doesn't give a value of 1 for the expression ##\frac{2x}{1 - x^2}\sqrt 3##.
I mean ##\frac{2x}{1 - x^2}\sqrt 3=1##
 
littlemathquark said:
TL;DR Summary: İf ##-1+\dfrac{2}{1-x}=\sqrt 3## then find ##\dfrac{2x}{1-x^2}\cdot \sqrt 3##

But I need another way.
a) Why?
b) Explain the techniques that qualify for "another".
 
fresh_42 said:
a) Why?
b) Explain the techniques that qualify for "another".
a) Just curious
b) I don't know the techniques that qualify as "another"
 
littlemathquark said:
a) Just curious
b) I don't know the techniques that qualify as "another"
So let me make this clear: you have a precise definition of the value ##x## and want to find a value ##g(x)## without using ##x## and no idea what is allowed to be used instead?
 
fresh_42 said:
So let me make this clear: you have a precise definition of the value ##x## and want to find a value ##g(x)## without using ##x## and no idea what is allowed to be used instead?
I want to find value of g(x) using/manupulating given equation( as I said I found it using trigonometry but I wonder other ways) . I assume that not known precise value of x.
 
littlemathquark said:
I can solve it with two ways:
1) directly by finding ##x##
2) Let ##x=\tan\alpha## and go on and answer is ##1.##
I get a solution for x that checks out. If I substitute this value into ##\frac{2x}{1 - x^2}\sqrt 3##, I don't get a result of 1 for that expression.

What did you get for x?
If you replace x in the expression above by ##\tan(\alpha)##, the expression becomes ##\frac{2\tan(\alpha)}{1 - \tan^2(\alpha)}\sqrt 3 = \sqrt 3 \tan(2\alpha)##. How does that simplify to 1?
 
Mark44 said:
I get a solution for x that checks out. If I substitute this value into ##\frac{2x}{1 - x^2}\sqrt 3##, I don't get a result of 1 for that expression.

What did you get for x?
If you replace x in the expression above by ##\tan(\alpha)##, the expression becomes ##\frac{2\tan(\alpha)}{1 - \tan^2(\alpha)}\sqrt 3 = \sqrt 3 \tan(2\alpha)##. How does that simplify to 1?
I found ##x=2-\sqrt 3## and evaluated it in ##(2x/(1-x^2)).\sqrt 3##
 
Last edited:
  • #10
littlemathquark said:
I found ##x=2-\sqrt 3## and evaluated it ##(2x/(1-x^2)).\sqrt 3##
##(x+1)/(1-x)=\sqrt 3## and Let ##x=\tan\alpha## so ##(1+\tan\alpha)/(1-\tan\alpha)=\sqrt 3## and ##1=\tan 45##
##\tan(45+\alpha)=\sqrt 3=\tan 60##
##\alpha=15^{\circ} ##
 
  • #11
littlemathquark said:
I found ##x=2-\sqrt 3## and evaluated it in ##(2x/(1-x^2)).\sqrt 3##
Rechecking my work, I get the same value for x as you did.
 
  • #12
\frac{2x}{1-x^2}=\frac{1}{1- x}-\frac{1}{1+x}with\frac{1+x}{1-x}=\sqrt{3}seems useful.
 
  • #13
anuttarasammyak said:
\frac{2x}{1-x^2}=\frac{1}{1- x}-\frac{1}{1+x}with\frac{1+x}{1-x}=\sqrt{3}seems useful.
I can't use your hint. Can you give me more detail?
 
  • #14
\frac{2x}{1-x^2}=\frac{1}{1- x}-\frac{1}{1+x}=(1-\frac{1}{\sqrt{3}})\frac{1}{1-x}=(1-\frac{1}{\sqrt{3}})\frac{\sqrt{3}+1}{2}=\frac{1}{\sqrt{3}}
 
  • Like
Likes littlemathquark
  • #15
Thank you. Now, I have three ways for solution the problem.
 
  • #16
Geometry of
2-\sqrt{3}=tan \frac{\pi}{12}
HI-20250315_11173985.jpg
 
  • #17
anuttarasammyak said:
Geometry of
2-\sqrt{3}=tan \frac{\pi}{12}
View attachment 358523
littlemathquark said:
Thank you. Now, I have three ways for solution the problem.
Fourth solution:
##\dfrac{1+x}{1-x}=\sqrt3## so ##(1+x,1-x)=(\sqrt3k,k)## and ##k=\sqrt3-1##

##2x=(1+x)-(1-x)## and ##1-x^2=(1+x)(1-x)## so ##\dfrac{(\sqrt3-1)k}{\sqrt3k\cdot k}\cdot \sqrt 3=1##
 
  • #18
\frac{2\sqrt{3}x}{1-x^2}:=A
Ax^2+2\sqrt{3}x-A=0
x=\frac{-\sqrt{3}\pm\sqrt{A^2+3}}{A}=2-\sqrt{3}
We observe
A=1 with + sign satisfies the relation.
 
Last edited:
  • Like
Likes littlemathquark
Back
Top