Question

If 2(4x + 3)/(x - 3)(x + 7) = a/x - 3 + b/x + 7, find the values of a and b.

Answer 1

Explanation:

look this solution

Answer 2

`a=3` and
`b=5`.

Explanation:

So I believe the problem is this:

`(2(4x+3))/(x-3)(x+7)}=(a)/(x-3)+(b)/(x+7)`

where we are asked to find values for
`a` and
`b` such that the equation holds for any
`x` in the equation's domain.

So I'm actually going to get rid of any domain restrictions by multiplying both sides by (x-3)(x+7).

In other words this will clear the fractions.

`(2(4x+3))/(x-3)(x+7)}\cdot(x-3)(x+7)=(a)/(x-3)\cdot(x-3)(x+7)+(b)/(x+7)(x-3)(x+7)`

`2(4x+3)=a(x+7)+b(x-3)`

As you can see there was some cancellation.

I'm going to plug in -7 for x because x+7 becomes 0 then.

`2(4\cdot -7+3)=a(-7+7)+b(-7-3)`

`2(-28+3)=a(0)+b(-10)`

`2(-25)=0-10b`

`-50=-10b`

Divide both sides by -10:

`(-50)/(-10)=b`

`5=b`

Now we have:

`2(4x+3)=a(x+7)+b(x-3)` with
`b=5`

I notice that x-3 is 0 when x=3. So I'm going to replace x with 3.

`2(4\cdot 3+3)=a(3+7)+b(3-3)`

`2(12+3)=a(10)+b(0)`

`2(15)=10a+0`

`30=10a`

Divide both sides by 10:

`(30)/(10)=a`

`3=a`

So
`a=3` and
`b=5`.