If the sum of two numbers is $22$ and the sum of their squares is $404$, then the product of the numbers is

### Concept & Formula This problem tests your familiarity with the fundamental algebraic expansion of a perfect square. The product of two numbers can be isolated if their sum and the sum of their squares are known. $$(x + y)^2 = x^2 + y^2 + 2xy$$ ### Step-by-Step Solution * **Given:** The sum of the two numbers is $22$, so $x + y = 22$. * **Given:** The sum of their squares is $404$, so $x^2 + y^2 = 404$. * We need to find the product, which is $xy$. * Substitute the given values into the perfect square formula: $$(22)^2 = 404 + 2xy$$ * Calculate the square of $22$: $$484 = 404 + 2xy$$ * Rearrange to isolate the $2xy$ term: $$2xy = 484 - 404$$ $$2xy = 80$$ * Divide by $2$ to find the product: $$xy = 40$$ ### Exam Strategy & Shortcut Memorize squares up to $30$ perfectly. If you instantly know $22^2 = 484$, this entire problem can be solved mentally in under $10$ seconds. Mental track: $484 - 404 = 80$. Half of $80$ is $40$. Mark the answer and move on immediately. ### Common Pitfall The most frequent mistake here is forgetting to divide the final difference by $2$. Students often stop at $80$ (which is $2xy$) and incorrectly select option (c). Always remember that the formula generates $2xy$, not $xy$. ### Final Answer **Therefore, the correct answer is 40.**