Question

Find the value of A+B

5/x²+6x+8 = A/x+2 + B/x+4

Answer 1

Explanation:

`\frac{5}{ {x }^(2) + 6x + 8} = (a)/(x + 2) + (b)/(x + 4) \\ factorizing \: the \: denominator \: of \: the \: first \: term \\ {x}^(2) + 6x + 8 \\ {x}^(2) + 4x + 2x + 8 \\ ( {x}^(2) + 4x)( + 2x + 8) \\ x(x + 4) + 2(x + 4) \\( x + 2)(x + 4) \\ \\ \\ \\ (5)/((x + 2)(x + 4)) = (a)/(x + 2) + (b)/(x + 4 ) \\ multiplying \: by \: the \: lcm = (x + 2)(x + 4) \\ 5 = a(x + 4) + b(x + 2) \\ put \: x = - 4 \\ 5 = a( - 4 + 4) + b( - 4 + 2) \\ 5 = a(0) + b( - 2) \\ 5 = 0 - 2b \\ 5 = - 2b \\ b = (5)/( - 2) \\ b = - 2 \ (1)/(2) \\ substituting \: b = - 2 (1)/(2) and \: put \: x = - 2 \\ 5 = a( - 2 + 4) + b( - 2 + 2) \\ 5 = a(2) + b(0) \\ 5 = 2a \\ a = 2 (1)/(2) \\ a + b = 2 (1)/(2) + ( - 2 (1)/(2) ) \\ 2 (1)/(2) - 2 (1)/(2) \\ a + b = 0`