What is the correct form of the partial fraction decomposition for the expression 7x+18/x^2+9x

x ² + 9x = x (x + 9), so the partial fraction decomposition would be

(7x + 18) / (x ² + 9x) = a/x + b/(x + 9)

where a, b are some constants. To find them, multiply both sides by x ² + 9x :

7x + 18 = a (x + 9) + bx

Solve for a and b :

7x + 18 = (a + b) x + 9a

[ a + b = 7

[ 9a = 18

→   a = 2, b = 5

So

(7x + 18) / (x ² + 9x) = 2/x + 5/(x + 9)

its totaly b

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