If frac{13x+43}{2x^2+17x+30} = frac{A}{2x+5} | vedclass

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Question

(C) Given the partial fraction decomposition: $\frac{13x+43}{2x^2+17x+30} = \frac{A}{2x+5} + \frac{B}{x+6}$ Combining the terms on the right side: $\f

Vedclass · Accepted Answer

$34$

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(C) Given the partial fraction decomposition: $\frac{13x+43}{2x^2+17x+30} = \frac{A}{2x+5} + \frac{B}{x+6}$ Combining the terms on the right side: $\frac{13x+43}{2x^2+17x+30} = \frac{A(x+6) + B(2x+5)}{(2x+5)(x+6)}$ Since the denominators are equal,we equate the numerators: $13x + 43 = A(x+6) + B(2x+5)$ $13x + 43 = (A+2B)x + (6A+5B)$ Comparing the coefficients of $x$ and the constant terms: $A + 2B = 13$  $(i)$ $6A + 5B = 43$  $(ii)$ Multiply equation $(i)$ by $6$: $6A + 12B = 78$  $(iii)$ Subtract equation $(ii)$ from $(iii)$: $(6A + 12B) - (6A + 5B) = 78 - 43$ $7B = 35 \Rightarrow B = 5$ Substitute $B = 5$ into equation $(i)$: $A + 2(5) = 13$ $\Rightarrow A + 10 = 13$ $\Rightarrow A = 3$ Finally,calculate $A^2 + B^2$: $A^2 + B^2 = 3^2 + 5^2 = 9 + 25 = 34$

If $\frac{13x+43}{2x^2+17x+30} = \frac{A}{2x+5} + \frac{B}{x+6}$,then $A^2+B^2=$

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