Working With Algebraic Expressions

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[tygrataps]

I was thinking about flipping the fraction over instead of dividing by x^2. Close but not quite the same.

[আল_ইশিগামি_সেনকু]

Given that,

X+1/x = 3

=> X^2 + 1 = 3x

Now,

3x^2/(x^4 -x^2+1)

= 3x^2/[(x^2 +1)^2 -3x^2]

=3x^2/9x^2 - 3x^2

=3x^2÷6x^2 = 1/2

[chaosredefined]

x + 1/x = 3

Square both sides
x^2 + 2(x)(1/x) + 1/x^2 = 9
x^2 + 2 + 1/x^2 = 9

Subtract 3 from both sides
x^2 - 1 + 1/x^2 = 6

Multiply through by x^2
x^4 - x^2 + 1 = 6x^2

Take the reciprocal of both sides
1/(x^4 - x^2 + 1) = 1/(6x^2)

Multiply both sides by 3x^2
(3x^2)/(x^4 - x^2 + 1) = (3x^2)/(6x^2)
(3x^2)/(x^4 - x^2 + 1) = 1/2

[josperez]

x + 1/x = 3 -> (x^2 +1)/x = 3 -> x/(x^2+1) = 1/3 -> x^2/(x^2+1)^2 = 1/9 -> 3x^2/(x^2+1)^2 = 1/3 -> 3x^2/(x^4 + 1 + 2x^2) = 1/3 -> 3x^2/(x^4 + 1 - x^2 + 3x^2) = 1/3 -> (x^4 + 1 - x^2 + 3x^2)/(3x^2) = 3 -> (x^4 + 1 - x^2)/(3x^2) + 3x^2/3x^2 = 3 -> (x^4 + 1 - x^2)/(3x^2) + 1 = 3 -> (x^4 + 1 - x^2)/(3x^2) = 2 -> 3x^2/(x^4 + 1 - x^2) = 1/2

[davidbrisbane]

Observe that (x⁴ - x² + 1)/x² = x² - 1 + 1/x²

Now x + 1/x = 3
So, (x + 1/x)² = x² + 2 + 1/x²
= 3² = 9
So, x² - 1 + 1/x² = 9 - 3 = 6.

So, x²/(x⁴ - x² + 1) = 1/6
So, 3x²/(x⁴ - x² + 1) = 1/2

[SidneiMV]

x + 1/x = 3

E = 3x²/(x⁴ - x² + 1)
E = 3/(x² + 1/x² - 1)

x + 1/x = 3
x² + 1/x² = 7

E = 3/(7 - 1)
E = 1/2