This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here's the step-by-step solution: The given expression is: (m+b)/(m-b) + (m-b)/(m+b) - (m^2+b^2)/(m^2-b^2) Step 1: Find a common denominator for the first two terms. The common denominator is (m-b)(m+b) = m^2-b^2. ((m+b)^2)/((m-b)(m+b)) + ((m-b)^2)/((m+b)(m-b)) - (m^2+b^2)/(m^2-b^2) ((m+b)^2 + (m-b)^2)/(m^2-b^2) - (m^2+b^2)/(m^2-b^2) Step 2: Expand the squares in the numerator. Recall that (A+B)^2 = A^2 + 2AB + B^2 and (A-B)^2 = A^2 - 2AB + B^2. ((m^2 + 2mb + b^2) + (m^2 - 2mb + b^2))/(m^2-b^2) - (m^2+b^2)/(m^2-b^2) Step 3: Combine like terms in the numerator. The 2mb and -2mb terms cancel out. (m^2 + m^2 + b^2 + b^2)/(m^2-b^2) - (m^2+b^2)/(m^2-b^2) (2m^2 + 2b^2)/(m^2-b^2) - (m^2+b^2)/(m^2-b^2) Step 4: Since both terms now have the same denominator, combine their numerators. ((2m^2 + 2b^2) - (m^2+b^2))/(m^2-b^2) Step 5: Distribute the negative sign in the numerator and combine like terms. (2m^2 + 2b^2 - m^2 - b^2)/(m^2-b^2) ((2m^2 - m^2) + (2b^2 - b^2))/(m^2-b^2) (m^2 + b^2)/(m^2-b^2) The final answer is (m^2+b^2)/(m^2-b^2).