Question

What is $a$ square plus $b$ square?

Answer 1

Different formulas for $a^2+b^2$:

Using algebraic properties:

1. $\left(a^2+b^2\right)={\left(a+b\right)}^2-2ab$ $\left[\because {\left(a+b\right)}^2=a^2+b^2+2ab\right]$
2. $\left(a^2+b^2\right)={\left(a-b\right)}^2+2ab$ $\left[\because {\left(a-b\right)}^2=a^2+b^2-2ab\right]$

Using complex numbers:

${\mathit{a}}^{\mathbf{2}}\mathbf{}\mathbf{+}\mathbf{}{\mathit{b}}^{\mathbf{2}}\mathbf{}\mathbf{=}\mathbf{}\left(a+ib\right)\left(a-ib\right)$

By Pythagoras theorem:

If $a$ is base and $b$ is perpendicular to a right-angled triangle, then ${\mathit{a}}^{\mathbf{2}}\mathbf{}\mathbf{+}\mathbf{}{\mathit{b}}^{\mathbf{2}}\mathbf{}\mathbf{=}\mathbf{}{\mathit{c}}^{\mathbf{2}}$. where $c$ is the length of hypotenuse.