Evaluate $475 \times 475 + 125 \times 125$

### Concept & Formula This expression represents the sum of two squares, $a^2 + b^2$. We can manipulate the standard algebraic identities for $(a+b)^2$ and $(a-b)^2$ to create a simpler calculation formula: $$ a^2 + b^2 = \frac{1}{2} [ (a + b)^2 + (a - b)^2 ] $$ ### Step-by-Step Solution * Let $a = 475$ and $b = 125$. * We need to find the value of $a^2 + b^2$. * Using the derived formula, substitute the values: $$ (475)^2 + (125)^2 = \frac{1}{2} [ (475 + 125)^2 + (475 - 125)^2 ] $$ * Calculate the sum and difference inside the brackets: $$ 475 + 125 = 600 $$ $$ 475 - 125 = 350 $$ * Substitute these back into the equation: $$ = \frac{1}{2} [ (600)^2 + (350)^2 ] $$ * Square the simpler numbers: $$ = \frac{1}{2} [ 360000 + 122500 ] $$ * Add the inner values and divide by 2: $$ = \frac{1}{2} \times 482500 = 241250 $$ ### Exam Strategy & Shortcut Whenever you see a pattern of $a^2 + b^2$ where $a$ and $b$ sum to a clean round number (like $475 + 125 = 600$), this specific identity is the fastest path. It replaces nasty 3-digit multiplications with much simpler mental math. ### Common Pitfall The most common mistake is correctly calculating the sum of the squares inside the bracket (482500) but forgetting to multiply by the $\frac{1}{2}$ at the very end. Test setters always include this halfway value as an option. ### Final Answer Therefore, the correct answer is **241250**.