Question: Determine the constants A and B that make the equation an identity. \[ \frac{36}{(x - 6)(x + 6)}…

Determine the constants A and B that make the equation an identity.

\[ \frac{36}{(x - 6)(x + 6)} = \frac{A}{x - 6} + \frac{B}{x + 6} \]

\[ \frac{36}{(x - 6)(x + 6)} = \frac{\boxed{}}{x - 6} + \frac{\boxed{}}{x + 6} \]

Solution

To solve the problem, we need to determine constants \( A \) and \( B \) such that the equation is an identity: \[ \frac{36}{(x-6)(x+6)} = \frac{A}{x-6} + \frac{B}{x+6} \] To combine the right side into a single fraction, we have: \[ \frac{A}{x-6} + \frac{B}{x+6} = \frac{A(x+6) + B(x-6)}{(x-6)(x+6)} \] This means: \[ A(x+6) + B(x-6) = 36 \] Expanding both terms: \[ Ax + 6A + Bx - 6B = 36 \] Combine like terms: \[ (A + B)x + (6A - 6B) = 36 \] For the identity to hold for all \( x \), the coefficients of corresponding terms must be equal. So, we set up the following system of equations: \[ \begin{cases} A + B = 0 & \\ 6A - 6B = 36 & \end{cases} \] From \( A + B = 0 \), we solve for \( B \): \[ B = -A \] Substitute \( B = -A \) into \( 6A - 6B = 36 \): \[ 6A - 6(-A) = 36 \] Simplify: \[ 6A + 6A = 36 \] \[ 12A = 36 \] Divide by 12: \[ A = 3 \] Using \( B = -A \): \[ B = -3 \] Therefore, the constants are \( A = 3 \) and \( B = -3 \).