Four students, Michal, Bob, Pavel, and Marek, solved the equation: $$ (x+1)^2+x(x−1)=2(x−1)^2 $$ Which of them proceeded correctly in simplifying the equation?
Michal: $$\begin{aligned} x^2+1+x^2-x&=2(x^2-1) \cr 2x^2-x+1&=2x^2-2 \end{aligned}$$
Bob: $$\begin{aligned} x^2+2x+1+x^2-x&=(2x-2)^2 \cr 2x^2+x+1&=4x^2-8x+4 \end{aligned}$$
Pavel: $$\begin{aligned} x^2+2x+1+x^2-1x&=2(x^2-2x+1) \cr 2x^2+x+1&=2x^2-4x+2 \end{aligned}$$
Marek: $$\begin{aligned} x^2+x+1+x^2−x&=2(x^2-x+1) \cr 2x^2+1&=2x^2-2x+2 \end{aligned}$$
Pavel
Bob
Marek
Michal
None of them
Pavel correctly squared $(x+1)^2=x^2+2x+1$, $(x−1)^2=x^2-2x+1$. Unlike Bob, he realized that exponentiation has higher precedence than multiplication, so he squared $(x-1)$ first and only then used the distributive property to remove the parentheses. Michal and Marek made an error when squaring $(x+1)$ and $(x-1)$.