Question

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

• A. 8
• B. 18
• C. 3
• D. 5

Hint:

For such problems, it’s helpful to equate the numerators after factoring the denominator, then solve the resulting system of equations.

Answer 1

The Correct Option is A

We need to find the value of \( A + B \). First, rewrite the given equation as: \[ \frac{13x+43}{2x^2 + 17x + 30} = \frac{A}{2x+5} + \frac{B}{x+6} \] The denominator on the left-hand side can be factored as: \[ 2x^2 + 17x + 30 = (2x+5)(x+6) \] Now, equate the two fractions: \[ \frac{13x + 43}{(2x+5)(x+6)} = \frac{A(x+6) + B(2x+5)}{(2x+5)(x+6)} \] Now, equate the numerators: \[ 13x + 43 = A(x+6) + B(2x+5) \] Expanding both sides: \[ 13x + 43 = A(x) + 6A + B(2x) + 5B \] \[ 13x + 43 = (A + 2B)x + (6A + 5B) \] By comparing the coefficients of \(x\) and the constant term, we get the system of equations: 1. \( A + 2B = 13 \) 2. \( 6A + 5B = 43 \)
Solving this system, we first multiply the first equation by 5: \[ 5A + 10B = 65 \] Now subtract the second equation from this: \[ (5A + 10B) - (6A + 5B) = 65 - 43 \] \[ -A + 5B = 22 \] \[ A = 5B - 22 \] Substitute this into the first equation: \[ (5B - 22) + 2B = 13 \] \[ 7B - 22 = 13 \] \[ 7B = 35 \] \[ B = 5 \] Now, substitute \(B = 5\) into \(A = 5B - 22\): \[ A = 5(5) - 22 = 25 - 22 = 3 \] Thus, \(A = 3\) and \(B = 5\), so: \[ A + B = 3 + 5 = 8 \]