If $(a - b) = 4$ and $ab = 2$, then $(a^2 + b^2) = ?$

### Concept & Formula This problem requires evaluating the sum of squares using the given difference of the variables and their product. We use the fundamental algebraic identity for a squared difference. $$(a - b)^2 = a^2 + b^2 - 2ab$$ ### Step-by-Step Solution Given: $(a - b) = 4$ $ab = 2$ We need to find the value of $a^2 + b^2$. Take the standard identity: $(a - b)^2 = a^2 - 2ab + b^2$ Rearrange the equation to isolate the desired term $a^2 + b^2$: $a^2 + b^2 = (a - b)^2 + 2ab$ Substitute the given values into the rearranged equation: $a^2 + b^2 = (4)^2 + 2(2)$ $a^2 + b^2 = 16 + 4$ $a^2 + b^2 = 20$ ### Exam Strategy & Shortcut **Direct Substitution:** Memorize the rearranged forms of algebraic identities. Knowing $a^2 + b^2 = (a-b)^2 + 2ab$ allows you to solve this mentally in seconds without writing down the intermediate steps. Just think: "$4^2 = 16$, plus double of 2 is 4, $16 + 4 = 20$." ### Common Pitfall Students frequently use the wrong sign when rearranging the formula. Because they start with $(a-b)^2$, they mistakenly subtract $2ab$ instead of adding it, leading to $16 - 4 = 12$. Always double-check your sign inversions when moving terms across the equals sign. ### Final Answer Therefore, the correct answer is **20**.