If $x + y = 15$ and $xy = 56$, then what is the value of $x^2 + y^2$?

### Concept & Formula This problem utilizes standard algebraic identities. The target expression $x^2 + y^2$ can be directly derived from the square of the sum of the variables. $$(x + y)^2 = x^2 + y^2 + 2xy$$ ### Step-by-Step Solution Given: $x + y = 15$ $xy = 56$ We need to find $x^2 + y^2$. Using the algebraic identity: $(x + y)^2 = x^2 + y^2 + 2xy$ Rearrange the formula to solve for $x^2 + y^2$: $x^2 + y^2 = (x + y)^2 - 2xy$ Substitute the given values into the equation: $x^2 + y^2 = (15)^2 - 2(56)$ $x^2 + y^2 = 225 - 112$ $x^2 + y^2 = 113$ ### Exam Strategy & Shortcut **Number Guessing (Factorization):** For problems with small, neat integers, try to guess the numbers logically. We need two numbers that multiply to 56 and add to 15. Think of the factors of 56: $1 \times 56$, $2 \times 28$, $4 \times 14$, $7 \times 8$. Notice that $7 + 8 = 15$. So, the numbers are 7 and 8. Now, calculate $x^2 + y^2 = 7^2 + 8^2 = 49 + 64 = 113$. This bypasses algebraic formulas entirely! ### Common Pitfall A common error is confusing the identity and calculating $(x+y)^2 + 2xy$ instead of subtracting $2xy$, leading to $225 + 112 = 337$, which usually throws students off when it isn't an option. ### Final Answer Therefore, the correct answer is **113**.