Find the numbers a and b from in relation (3x+1)/(x^2-4)=a/(x-2)+b/(x+2)

To determine a and b, we'll have to calculate the least
common denominator of the 3 ratios.

LCD = (x-2)(x+2) = x^2
- 4

LCD =  x^2 - 4

Now, we'll
multiply the first ratio from the right side by x+2 and the second ratio by x-2. The
ratio from the left side has the denominator x^2 - 4, so it won't be
multiplied.

We'll re-write the equation, all 3 quotients
having the denominator x^2 - 4.

3x + 1 = a(x+2) +
b(x-2)

We'll remove the
brackets:

3x + 1 = ax + 2a + bx -
2b

We'll combine the terms from the right side with respect
to x:

3x + 1 = x(a + b) + 2a -
2b

The expressions from both sides are equals if the
correspondent coefficients are equal:

The coefficient of x
from the left side has to be equal to the coefficient of x, from the right
side:

a + b = 3 (1)

2a - 2b =
1 (2)

We'll multiply (1) by 2 and we'll
get:

2a + 2b = 6 (3)

We'll add
(3)+(2):

2a - 2b + 2a + 2b = 1 +
6

We'll combine and eliminate like
terms:

4a = 7

a
= 7/4

We'll substitute a in
(1):

7/4 + b = 3

b = 3 -
7/4

b =
(12-7)/4

b =
5/4