2(4x+3) a b

Question

----------- ≡ (identically equal to) ------ + ------
(x-3)(x+7) x-3 x+7

(2(4x+3))/((x-3)(x+7)) ≡ a/(x-3) + b/(x+7)

Answer 1

Sorry, the spaces messed up, but the question/equation is on the bottom.

Answer 2

Looks like you are studying "partial fractions"

(2(4x+3))/((x-3)(x+7)) ≡ a/(x-3) + b/(x+7)
(8x + 6)/((x-3)(x+7)) ≡ (a(x+7) + b(x-3) )/(((x-3)(x+7))
multiply by (x-3)(x+7)

8x + 6 ≡ a(x+7) + b(x-3)

this must be true for all values of x
let x = 3, ---> 24+6 = 10a + 0
a = 3
let x = -7, ---> -56+6 = 0 - 10b
b = 5

you might have learned to expand 8x + 6 ≡ a(x+7) + b(x-3) and then comparing like terms,

8x + 6 = ax + 7a + bx - 3b
8x + 6 = x(a+b) + 7a-3b
then a + b = 8 ---> a = -b+8
and 7a - 3b = 6

by substitution:
7(-b+8) - 3b = 6
-7b + 56 - 3b = 6
-10b = -50
b = 5 , then a = -5+8 = 3

I find the method I used much simpler.

Answer 3

Thank you very much.