If $a + b = 7$ and $ab = 12$, what is the value of $a^{2} + b^{2}$?

Use the identity $(a + b)^{2} = a^{2} + 2ab + b^{2}$, which rearranges to $a^{2} + b^{2} = (a + b)^{2} - 2ab$. $a^{2} + b^{2} = 7^{2} - 2 \cdot 12 = 49 - 24 = 25$. Check: $a$ and $b$ satisfy $t^{2} - 7t + 12 = 0$, giving $t = 3, 4$. Then $a^{2} + b^{2} = 9 + 16 = 25$ ✓. - Trap C ($49$) returns $(a+b)^2$ without subtracting. - Trap D ($73 = 49 + 24$) adds instead of subtracting.