Two numbers differ by 5. If their product is 336, then the sum of the two numbers is

## Concept & Formula When you are given the difference and the product of two numbers and need to find their sum, you should use the core algebraic identity relating sum, difference, and product. This avoids finding the individual numbers entirely. $$ (x + y)^2 = (x - y)^2 + 4xy $$ ## Step-by-Step Solution * **Given:** The difference $(x - y) = 5$. The product $xy = 336$. * **Calculation / Deduction:** * We need to find the sum $(x + y)$. * Substitute the known values into the algebraic identity: * $(x + y)^2 = (5)^2 + 4(336)$ * Calculate the components: * $(x + y)^2 = 25 + 1344$ * $(x + y)^2 = 1369$ * Take the square root of both sides to find the sum: * $x + y = \sqrt{1369}$ * $x + y = 37$ ## Exam Strategy & Shortcut **Unit Digit Method:** You need the square root of $1369$. Look at the last digit, which is $9$. A perfect square ends in $9$ only if its square root ends in $3$ or $7$ (since $3 \times 3 = 9$ and $7 \times 7 = 49$). Look at the options. The only options ending in $3$ or $7$ is $37$. You don't even need to fully calculate $\sqrt{1369}$! ## Common Pitfall Students often attempt to set up a quadratic equation $x(x - 5) = 336 \Rightarrow x^2 - 5x - 336 = 0$. Factoring large numbers like $336$ under pressure is highly prone to errors and takes too much time. Use the algebraic identity instead. ## Final Answer **Therefore, the correct answer is 37.**