solving an equation?

Question

1
____
x + 2

+

2
____
x-2

=

1
____
x^2 + 4

Answer 1

.. 1 ....... 2 ........ 1
------ + ------ = ------
x + 2 .. x - 2 .. x² + 4

Get a common denominator on the left side by multiplying the first fraction by (x-2)/(x-2) and the second fraction by (x+2)/(x+2).

x - 2 .. 2(x+2).... 1
------ + -------- = --------
x² - 4 .. x² - 4 .... x² + 4

Now combine the fractions on the left:
x - 2 + 2x + 4 ...... 1
----------------- = --------
..... x² - 4 .......... x² + 4

Simplify the numerator:
Now combine the fractions on the left:
3x + 2 ....... 1
--------- = --------
x² - 4 ......x² + 4

Now cross multiply:
(3x + 2)(x² + 4) = x² - 4

Multiply through:
3x^3 + 2x² + 12x + 8 = x² - 4

Put everything on the left:
3x^3 + x² + 12x + 12 = 0

Now solve this for x and you are done...

However, I suspect that you have a typo in your equation and meant to have x² - 4 as the denominator on the right. If so, then you could multiply by x² - 4 and have
3x + 2 = 1
3x = -1
x = -1/3

Answer 2

Multiply the first figure by (x-2)/(x-2) and the second by (x+2)/(x+2)

This makes denominators equal on both sides of equal signs...now you have (x-2) + 2(x+2) = 1

Does that help?

Answer 3

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

[3x +2]/(x^2 + 4)

if x<>+/- 2

3x+ 2 = 1

x= -1/3