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Problem: Partial Fraction (5x-4)/(x^2-x-2) [ Calculator, Method and examples ]

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

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

2. Partial fraction 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

=>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 fraction
((5x-4))/((x+1)(x-2))=(3)/(x+1)+(2)/(x-2)

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