Question

Question #c311f

Answer 1

`A/B = 5/3`

We can find exact values of `A` and `B`:
`A=3/19`, `B=5/19`

Explanation:

Given: `1/[(3-5x)(2+3x)] = A/(3-5x) + B/(2+3x)`

Bring the right side to common denominator:
`A/(3-5x) + B/(2+3x) = [A(2+3x)+B(3-5x)]/[(3-5x)(2+3x)]`

Compare this with a given equality.
Obviously,
`1 = A(2+3x)+B(3-5x)` for any `x`.
Equivalently,
`(3A-5B)x+2A+3B-1=0`

Since this equality should hold for all `x`, the coefficient at `x` must be equal to zero, that is, `3A-5B=0` and the free member should also be equal to zero, that is, `2A+3B-1=0`.

From the first equality, we conclude that `A/B = 5/3`.
Using this ratio and the second equality, we can find `A` and `B`:
`2*5/3B+3B-1=0`
`10B+9B-3=0`
`B=3/19`
`A=5/19`

CHECK:
`5/19(2+3x)+3/19(3-5x) = (10+15x+9-15x)/19 = 1`
as it should.