Squares Squared

[BigGlenntheHeavy]

Show that the product $\displaystyle (a^2+b^2)(c^2+d^2)$ can be written as the sum of two squares.

 

[takariendire]

Hi,

I replied to your question by video :

http://www.blueteach.com/v/cfbR49l0R2o1VfOcpH

cheers !

 

[Denis]

"a rien dire" is French for "nothing to say" :wink:

 

[soroban]

Hello, "BigGlenntheHeavy!

This requires some Olympic-level gymnastics . . .

$\displaystyle \text{Show that the product }\,(a^2+b^2)(c^2+d^2)\,\text{ can be written as the sum of two squares.}$

\(\displaystyle \begin{array}{ccccc}\text{We have:} &a^2c^2 + a^2d^2 + b^2c^2 + b^2d^2 \\ \text{Regroup terms:} & (a^2c^2 + b^2d^2) + (a^2d^2 + b^2c^2) \\ \text{Add and subtract }2abcd\!:} & (a^2c^2 + 2abcd + b^2d^2) + (a^2d^2 - 2abcd + b^2c^2) \\ \text{Factor:} & (ac + bd)^2 + (ad - bc)^2 \end{array}\)


\(\displaystyle \begin{array}{ccccc}\text{We have:} &a^2c^2 + a^2d^2 + b^2c^2 + b^2d^2 \\ \text{Regroup terms:} & (a^2c^2 + b^2d^2) + (a^2d^2 + b^2c^2) \\ \text{Subtract and add }2abcd\!:} & (a^2c^2 - 2abcd + b^2d^2) + (a^2d^2 + 2abcd + b^2c^2) \\ \text{Factor:} & (ac - bd)^2 + (ad + bc)^2 \end{array}\)


$\displaystyle \text{Therefore, there are }two\text{ solutions:}$

.\(\displaystyle (a^2+b^2)(c^2+d^2) \;=\;\begin{array}{cc} (ac+bd)^2 + (ad-bc)^2 \\ \\[-3mm] (ac-bd)^2 + (ad+bc)^2 \end{array}\)