Question

What is the formula of $x^2+y^2$?

Answer 1

Write the formula for $x^2+y^2$.

From${\left(x+y\right)}^2={\mathit{x}}^{\mathbf{2}}\mathbf{+}{\mathit{y}}^{\mathbf{2}}\mathbf{+}\mathbf{2}\mathit{x}\mathit{y}$

$\begin{array}{rcl}{\left(x+y\right)}^2& =& x^2+y^2+2xy\\ & \Rightarrow & x^2+y^2={\left(x+y\right)}^2-2xy\end{array}$

Also, From${\left(x-y\right)}^{\mathbf{2}}\mathbf{=}{\mathit{x}}^{\mathbf{2}}\mathbf{+}{\mathit{y}}^{\mathbf{2}}\mathbf{-}\mathbf{2}\mathit{x}\mathit{y}$

$\begin{array}{rcl}{\left(x-y\right)}^2& =& x^2+y^2-2xy\\ & \Rightarrow & x^2+y^2={\left(x-y\right)}^2+2xy\end{array}$

Hence,the formula is $x^2+y^2={\left(x+y\right)}^2-2xy$ or $x^2+y^2={\left(x-y\right)}^2+2xy$.