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

MagicKitten Nov 10, 2019

#1**+1 **

You sure you typed that correctly?

What do you mean by

"If \(x^2+\frac{1}{x^2}\)"

Is it equal to something..?

CalculatorUser Nov 10, 2019

#3**+2 **

Fixed

Working off of \(3x^2=x^4+1\)

We add 2x^{2} 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 \(\boxed{\frac{1}{5}}\)

.CalculatorUser Nov 10, 2019

edited by
CalculatorUser
Nov 10, 2019

edited by CalculatorUser Nov 10, 2019

edited by CalculatorUser Nov 10, 2019

edited by CalculatorUser Nov 10, 2019

edited by CalculatorUser Nov 10, 2019

#7**+1 **

I forgot to add 2x^{2} to BOTH sides.

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

That took me long. Phew.

CalculatorUser
Nov 10, 2019

#11**0 **

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

CalculatorUser
Nov 10, 2019

#13**+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)

Guest Nov 11, 2019