Solve the equation (x + 1)^2 + (x - 1)^2 = (x + 1)^2 + (1 - x)^2.

Step-by-step solution

1. 1

   Look for matching pieces before expanding. The term $(x+1)^2$ appears on both sides, so it can be subtracted away immediately. What remains is the much simpler question of whether $(x-1)^2=(1-x)^2$.

2. 2

   Use the fact that opposites square to the same value. $1-x=-(x-1)$, and squaring kills the sign: $(1-x)^2=\left(-(x-1)\right)^2=(-1)^2(x-1)^2=(x-1)^2$. So the two remaining terms are equal for every $x$.

3. 3

   Confirm by expanding both. $(x-1)^2=x^2-2x+1$ and $(1-x)^2=1-2x+x^2$; term by term these agree, which is the algebraic version of the same observation.

4. 4

   Conclude that the equation is an identity. After the cancellations the equation reads $0=0$, a statement with no variable in it that happens to be true. That is the signature of an identity: it holds for every value of the unknown.

5. 5

   State the solution set. Since no value of $x$ is excluded — there are no denominators or square roots to restrict the domain — the solution set is all real numbers, $x\in\mathbb{R}$.

6. 6

   Check two sample values. At $x=3$ both sides equal $16+4=20$; at $x=-7$ both sides equal $36+64=100$. Matching values at arbitrary test points is consistent with an identity rather than an equation with isolated roots.