Question

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

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

Hint:

In partial fractions, after taking LCM, compare the coefficients of like powers of \(x\) to find unknown constants.

Answer 1

The Correct Option is C

Step 1: Factor the denominator.
$2x^2+17x+30=(2x+5)(x+6)$, so $\dfrac{13x+43}{2x^2+17x+30}=\dfrac{A}{2x+5}+\dfrac{B}{x+6}$.
Step 2: Clear the fractions.
Multiplying through gives $13x+43=A(x+6)+B(2x+5)$.
Step 3: Find $B$ using a clever value.
Put $x=-6$: $13(-6)+43=-78+43=-35$ and the right side is $B(2(-6)+5)=B(-7)$, so $-7B=-35$, giving $B=5$.
Step 4: Find $A$ using another value.
Put $x=-\dfrac52$: $13\left(-\tfrac52\right)+43=-\tfrac{65}{2}+43=\tfrac{21}{2}$ and the right side is $A\left(-\tfrac52+6\right)=A\cdot\tfrac72$, so $\tfrac72 A=\tfrac{21}{2}$, giving $A=3$.
Step 5: Compute $A^2+B^2$.
$A^2+B^2=3^2+5^2=9+25=34$.
Step 6: Conclude.
This is option (3).
\[ \boxed{34} \]