What two rational expressions sum to 4x+1/x^2−4x−12?
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4x+1/x^2−4x−12 = __/__(x-6) + __/__(x+2)
+23252 (4x + 1) / (x2 - 4x - 12) = (4x + 1) / [ (x + 2)(x - 6) ]
(4x + 1) / [ (x + 2)(x - 6) ] = A(x + 2) / [ (x + 2)(x - 6) ] + B(x - 6) / [ (x + 2)(x - 6) ]
---> (4x + 1) = A(x + 2) + B(x - 6)
---> 4x + 1 = Ax + 2A + Bx - 6B
---> Ax + Bx = 4x and 2A - 6B = 1
---> A + B = 4
Combining the above two equations: 2A - 6B = 1 and A + B = 4
A + B = 4 ---> x6 ---> 6A + 6B = 24
2A - 6B = 1 ---> 2A - 6B = 1
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---> (adding) 8A = 25 ---> A = 25/8
Since A + B = 4 ---> B = 4 - B ---> B = 4 - 25/8 ---> B = 7/8
