The equation x² - 7x + c = 0 has two real solutions that differ by 3. What is the value of c?

Let the two solutions be $r$ and $r + 3$. For $x^2 - 7x + c = 0$, the sum of the roots equals $7$ (the negative of the $x$-coefficient), so $r + (r + 3) = 7$, giving $2r = 4$ and $r = 2$. The roots are 2 and 5. The product of the roots equals $c$, so $c = 2 \cdot 5 = 10$. Verify: $x^2 - 7x + 10 = (x-2)(x-5)$, and the roots differ by 3. Common wrong answers are 12, from using roots 3 and 4 (which sum to 7 but differ by only 1), and 7, from confusing $c$ with the sum of the roots.