Algebraic Expansion/Examples/(x + 1)^2

Example of Algebraic Expansion

The following is an algebraic expansion:

$\paren {x + 1}^2 = x^2 + 2 x + 1$

Proof 1

Use Square of Sum:

$\forall x, y \in \R: \paren {x + y}^2 = x^2 + 2 x y + y^2$

substituting $1$ for $y$.

$\blacksquare$

Proof 2

\(\ds \paren {x + 1}^2\)

\(=\)

\(\ds \paren {x + 1} \paren {x + 1}\)

\(\ds \)

\(=\)

\(\ds x \paren {x + 1} + 1 \paren {x + 1}\)

Distributive Property

\(\ds \)

\(=\)

\(\ds x^2 + 2 x + 1\)

Distributive Property

$\blacksquare$