If x + y = 7 and xy = 10, what is the value of x² + y²?

Use the identity $(x+y)^2 = x^2 + 2xy + y^2$, which rearranges to $x^2 + y^2 = (x+y)^2 - 2xy$. Substituting $x + y = 7$ and $xy = 10$ gives $49 - 20 = 29$. There is no need to find $x$ and $y$ individually, though they are 2 and 5, and indeed $4 + 25 = 29$. The most common wrong answer is 49, which comes from forgetting that squaring a sum adds the cross term $2xy$. Another is 39, from subtracting $xy$ once instead of twice, and a third is 69, from adding $2xy$ rather than subtracting it.