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Find Partial Fractions (5x-4)/(x^2-x-2)

Solution:
Your problem `->` Partial Fractions (5x-4)/(x^2-x-2)


1. Factors the denominator
`(5x-4)/(x^2-x-2)=(5x-4)/((x+1)(x-2))`

2. Partial fractions for each factors
`:. (5x-4)/((x+1)(x-2))=A/(x+1)+B/(x-2)`

3. Multiply through by the common denominator of `(x+1)(x-2)`

`:. 5x-4=A(x-2)+B(x+1)`

`:. 5x-4=Ax-2A+Bx+B`

4. Group the `x`-terms and the constant terms

`:. 5x-4=(A+B)x+(-2A+B)`

5. Coefficients of the two polynomials must be equal, so we get equations
`A+B=5`

`-2A+B=-4`

Solution of equations using Elimination method

Total Equations are `2`

`A+B=5 -> (1)`

`-2A+B=-4 -> (2)`



Select the equations `(1)` and `(2)`, and eliminate the variable `B`.

`A+B=5`` xx 1->````A``+``B``=``5```
`-2A+B=-4`` xx 1->``-``2A``+``B``=``-4```

```3A``=``9`` -> (3)`




Now use back substitution method
From (3)
`3A=9`

`=>3A=9`

`=>A=(9)/(3)=3`

From (1)
`A+B=5`

`=>(3)+B=5`

`=>B+3=5`

`=>B=5-3=2`

Solution using Elimination method.
`A = 3,B = 2`



After solving these equations, we get
`A=3,B=2`

Substitute these values in the original fractions
`((5x-4))/((x+1)(x-2))=(3)/(x+1)+(2)/(x-2)`






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