+142 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
+2862 You sure you typed that correctly?
What do you mean by
"If \(x^2+\frac{1}{x^2}\)"
Is it equal to something..?
+2862 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 \(\boxed{\frac{1}{5}}\)
+2862 I forgot to add 2x2 to BOTH sides.
don't worry, its fixed. The solution should be correct.
That took me long. Phew.
+2862 me because I attempted 4 different solutions and finally found the right one:
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)
