If $(a^2 - b^2) \div (a - b) = 25$, then $(a + b) = x$

## Concept & Formula This problem tests your recognition of the "Difference of Squares" algebraic formula, which is one of the most foundational identities in math. The formula is: $$a^2 - b^2 = (a - b)(a + b)$$ ## Step-by-Step Solution * **Given:** The equation provided is $(a^2 - b^2) \div (a - b) = 25$. Let the unknown target $(a + b)$ be $x$. * **Substitution:** Replace the numerator $a^2 - b^2$ with its expanded identity format. $$\frac{(a - b)(a + b)}{(a - b)} = 25$$ * **Simplification:** As long as $a \neq b$ (which implies $a - b \neq 0$), we can safely cancel out the common factor $(a - b)$ from both the numerator and the denominator. $$a + b = 25$$ * **Deduction:** The expression mathematically reduces directly to the value the question is asking for. $$x = 25$$ ## Exam Strategy & Shortcut This question shouldn't require any pen-and-paper calculation. The moment you see $(a^2 - b^2)$ divided by one of its factors (either $a+b$ or $a-b$), the answer is simply the *other* factor. Since it's set equal to 25, the other factor is exactly 25. Mark it and move on instantly. ## Common Pitfall Some students try to overcomplicate this by arbitrarily assuming values for $a$ and $b$ (e.g., trying to find perfect squares that subtract to a multiple of 25). While this might eventually work, it burns precious time. Rely strictly on the algebra here. ## Final Answer Therefore, the correct answer is 25.