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

### Concept & Formula This problem is a direct application of standard algebraic identities. The sum of squares can be derived algebraically using the expansion of the square of a sum. $$ (x + y)^2 = x^2 + y^2 + 2xy $$ ### Step-by-Step Solution * **Given:** $x + y = 15$ $xy = 56$ * **Calculation / Deduction:** Rearrange the algebraic identity to isolate the target expression, $x^2 + y^2$: $$ x^2 + y^2 = (x + y)^2 - 2xy $$ Substitute the given values into the formula: $$ x^2 + y^2 = (15)^2 - 2(56) $$ $$ x^2 + y^2 = 225 - 112 $$ $$ x^2 + y^2 = 113 $$ ### Exam Strategy & Shortcut **Number Guessing (Factor Pairing):** When the numbers are small, try to guess the integer values of $x$ and $y$ that satisfy both equations simultaneously. * Factors of $56$: $1 \times 56$, $2 \times 28$, $4 \times 14$, $7 \times 8$. * Check which pair adds up to $15$: $7 + 8 = 15$. * So, $x = 7$ and $y = 8$ (or vice versa). * Calculate $x^2 + y^2 = 7^2 + 8^2 = 49 + 64 = 113$. This completely bypasses algebraic expansion. ### Common Pitfall A frequent mistake is calculating $(x + y)^2$ and forgetting to subtract the $2xy$ term, leading a student to mistakenly assume the answer is $225$, or subtracting just $xy$ instead of $2xy$. ### Final Answer **Therefore, the correct answer is 113.**