The product of two numbers is 120 and the sum of their squares is 289. The sum of the numbers is

## Concept & Formula This problem connects the product of two numbers and the sum of their squares to their overall sum using the perfect square algebraic identity. $$ (x + y)^2 = x^2 + y^2 + 2xy $$ ## Step-by-Step Solution * **Given:** Let the two numbers be $x$ and $y$. Product: $xy = 120$ Sum of squares: $x^2 + y^2 = 289$ * **Calculation / Deduction:** * We need to find the sum of the numbers, $(x + y)$. * Write out the algebraic identity: * $$ (x + y)^2 = (x^2 + y^2) + 2(xy) $$ * Substitute the given values into the identity: * $$ (x + y)^2 = 289 + 2(120) $$ * $$ (x + y)^2 = 289 + 240 $$ * $$ (x + y)^2 = 529 $$ * Take the square root of both sides to find the sum: * $x + y = \sqrt{529}$ * $x + y = 23$ ## Exam Strategy & Shortcut **Pythagorean Triples / Factorization:** $289$ is $17^2$. This hints at the Pythagorean triple $(8, 15, 17)$ where $8^2 + 15^2 = 17^2$ ($64 + 225 = 289$). Let's check if these numbers satisfy the product condition: $8 \times 15 = 120$. It matches perfectly! The numbers are $8$ and $15$. Their sum is $8 + 15 = 23$. Recognizing common triples saves you from calculating the square root of $529$. ## Common Pitfall Forgetting the '2' in the $2xy$ term of the algebraic identity. If a student simply adds $289$ and $120$ they get $409$, which is not a perfect square and will lead to confusion and incorrect guessing. ## Final Answer **Therefore, the correct answer is 23.**