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If \(x^2 + \frac{1}{x^2}= 3\), what is the value of \(\frac{x^2}{(x^2+1)^2}\)?

 

Express your answer as a common fraction...

 

 

MY WORKIE:

 

\(x^2 + \frac{1}{x^2} = 3\)

 

\(\frac{x^4+1}{x^2}=\frac{3}{1}\)

 

 

\(3x^2 = x^4 + 1\)

 

\(\frac{x^2}{x^4+2x+1}=??????????\)

 

 

 

And that's all I've accomplished...

 

T-T

 Nov 10, 2019
edited by MagicKitten  Nov 10, 2019
 #1
avatar+2499 
+1

You sure you typed that correctly?
 

What do you mean by

 

"If \(x^2+\frac{1}{x^2}\)"

 

Is it equal to something..?

 Nov 10, 2019
 #2
avatar+141 
0

Sorry I just edited it....

 

It's equal to 3!

MagicKitten  Nov 10, 2019
 #3
avatar+2499 
+2

Fixed

 

Working off of \(3x^2=x^4+1\)

 

 

We add 2x2 to both sides

 

\(3x^2+2x^2=x^4+2x^2+1\)

 

Factor

\(5x^2=(x^2+1)^2\)

 

Divide both sides by \((x^2+1)^2\)

 

We get:

\(\frac{5x^2}{(x^2+1)^2}=1\)

 

Divide both sides by 5.

 

and you get your answer of \(\boxed{\frac{1}{5}}\)

.
 Nov 10, 2019
edited by CalculatorUser  Nov 10, 2019
edited by CalculatorUser  Nov 10, 2019
edited by CalculatorUser  Nov 10, 2019
 #4
avatar+141 
0

????????????????????????????????????????????????

MagicKitten  Nov 10, 2019
 #5
avatar+2499 
+1

fixed it.

CalculatorUser  Nov 10, 2019
 #6
avatar+141 
0

Wait- What?

MagicKitten  Nov 10, 2019
 #7
avatar+2499 
+1

I forgot to add 2x2 to BOTH sides.

 

don't worry, its fixed. The solution should be correct.

 

That took me long. Phew.

CalculatorUser  Nov 10, 2019
 #8
avatar+141 
0

But if you divide by 3, you get 1/3

MagicKitten  Nov 10, 2019
 #9
avatar+141 
0

oh you just edited it.

 

gotcha

MagicKitten  Nov 10, 2019
 #10
avatar+2499 
0

read my solution over again, lol.

CalculatorUser  Nov 10, 2019
 #11
avatar+2499 
0

me because I attempted 4 different solutions and finally found the right one:

 

CalculatorUser  Nov 10, 2019
 #12
avatar+141 
0

WHeLPpPpPPpp

MagicKitten  Nov 10, 2019
 #13
avatar
+1

I solved it by a different way,

Since x^2+1/x^2=3 

as you said, (x^4+1)/x^2=3 also. 

Now,

We want to find x^2/(x^2+1)^2

as you factorized it to (Notice you factorized it wrong by not putting power to "2x")

\(x^2/x^4+2x^2+1\)

Well we know that \(x^4+1/x^2=3\)

so solve for x^4+1 

\(x^4+1=3x^2\)

Now subsituite this into what we factorized we get:

\(x^2/3x^2+2x^2\)

Simplify,

\(x^2/5x^2\)

simplify more, we get it is equal to 1/5 (As the x's will cancel) 

 Nov 11, 2019
 #14
avatar+141 
0

THANK YOU

MagicKitten  Nov 12, 2019

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