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
To find a and b in (3x+1)/(x^2-4)=a/(x-2)+b/(x+2).
We notice that the denominators on both sides have LCM (x-2)(x+2) = x^2-4.
So multiply both sides by (x-2)(x+2) and get:
3x+1 = a(x+2) +b(x-2).
3x+1 = ax+2a+bx-2b
3x = (a+b)x +2(a-b).
Now equate the like terms on both sides:
(a+b)x = 3x . Or a+b = 3.....(1).
2(a-b) = 1. Or a-b = 1/2......(2).
Eq(1) +Eq(2) gives: 2a = 3+1/2. So a = 7/2*2 = 7/4 .
Eq(1)- eq(2) gives: 2b = 3-1/2 = 5/2. Or b = 5/4.
Therefore a = 7/4 and b = 5/2.