Question

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

• A. $\dfrac{22}{3}$
• B. $52$
• C. $34$
• D. $\dfrac{18}{5}$

Hint:

Use partial fractions and compare coefficients for solving rational expressions.

Answer 1

The Correct Option is C

Factor denominator: $2x^2 + 17x + 30 = (2x + 5)(x + 6)$
So write: $\dfrac{13x + 43}{(2x + 5)(x + 6)} = \dfrac{A}{2x + 5} + \dfrac{B}{x + 6}$
Multiply both sides: $13x + 43 = A(x + 6) + B(2x + 5)$
Expand: $Ax + 6A + 2Bx + 5B = (A + 2B)x + (6A + 5B)$
Match coefficients:
$A + 2B = 13$
$6A + 5B = 43$
Solving:
From first: $A = 13 - 2B$
Sub into second: $6(13 - 2B) + 5B = 43 \Rightarrow 78 - 12B + 5B = 43$
$\Rightarrow -7B = -35 \Rightarrow B = 5 \Rightarrow A = 3$
So $A^2 + B^2 = 3^2 + 5^2 = 9 + 25 = 34$