The product of two numbers is $45$ and the sum of their squares is $106$. The numbers are

### Concept & Strategy When given a system of non-linear equations involving products and sums of squares, substituting the options back into the conditions is often faster than solving the algebra algebraically. $$(a \times b) = \text{Product}$$ $$a^2 + b^2 = \text{Sum of Squares}$$ ### Step-by-Step Solution * **Given:** Let the two numbers be $a$ and $b$. * We know that $a \times b = 45$. * We know that $a^2 + b^2 = 106$. * Let's test the given options against these two conditions. * **Test Option (a) $3$ and $5$:** * Product: $3 \times 5 = 15 \neq 45$. (Incorrect) * **Test Option (b) $5$ and $9$:** * Product: $5 \times 9 = 45$. (Matches) * Sum of Squares: $5^2 + 9^2 = 25 + 81 = 106$. (Matches) * Since both conditions are perfectly satisfied, $5$ and $9$ are the correct numbers. ### Exam Strategy & Shortcut **Option Elimination:** For questions asking for "the numbers" when equations are provided, never solve the quadratic equations manually. Quickly scan the product condition first because factoring is easier mental math. Since $3 \times 5 = 15$ and $5 \times 19 = 95$, options (a) and (c) are instantly eliminated. Only (b) and (d) yield $45$. A quick glance at (d) shows $45^2$ is massive, leaving (b) as the only logical answer within $5$ seconds. ### Common Pitfall The most common mistake is attempting to solve $(a+b)^2 = a^2 + b^2 + 2ab$ to find $(a+b)$ and $(a-b)$, leading to a lengthy derivation. While algebraically correct, doing this under timed exam conditions eats up valuable minutes that could be saved by simply plugging in the choices. ### Final Answer **Therefore, the correct answer is 5 and 9.**