Expansions

Expansion is the process in which the contents of brackets are evaluated.

(a+b)^{2}=a^{2}+2ab+b^{2}

(a-b)^{2}=a^{2}-2ab+b^{2}

(a+b)^{2}+(a-b)^{2}=2(a^{2}+b^{2})

(a+b)^{2}-(a-b)^{2}=4ab

\left(a+\frac{1}{a}\right)^{2}=a^{2}+\frac{1}{a^{2}}+2 \Rightarrow a^{2}+\frac{1}{a^{2}}=\left(a+\frac{1}{a}\right)^{2}-2

\left(a-\frac{1}{a}\right)^{2}=a^{2}+\frac{1}{a^{2}}-2 \Rightarrow a^{2}+\frac{1}{a^{2}}=\left(a-\frac{1}{a}\right)^{2}+2

\left(a+\frac{1}{a}\right)^{2}+\left(a-\frac{1}{a}\right)^{2}=2\left(a^{2}+\frac{1}{a^{2}}\right)

\left(a+\frac{1}{a}\right)^{2}-\left(a-\frac{1}{a}\right)^{2}=4