Square of Sum/Algebraic Proof 1

Theorem

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

Proof

Follows from the distribution of multiplication over addition:

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

\(=\)

\(\ds \paren {x + y} \cdot \paren {x + y}\)

Definition of Square Function

\(\ds \)

\(=\)

\(\ds x \cdot \paren {x + y} + y \cdot \paren {x + y}\)

Real Multiplication Distributes over Addition

\(\ds \)

\(=\)

\(\ds x \cdot x + x \cdot y + y \cdot x + y \cdot y\)

Real Multiplication Distributes over Addition

\(\ds \)

\(=\)

\(\ds x \cdot x + x \cdot y + x \cdot y + y \cdot y\)

Real Multiplication is Commutative

\(\ds \)

\(=\)

\(\ds x \cdot x + 2 x y + y \cdot y\)

simplifying

\(\ds \)

\(=\)

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

Definition of Square Function

$\blacksquare$