BigGlenntheHeavy
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Show that the product \(\displaystyle (a^2+b^2)(c^2+d^2)\) can be written as the sum of two squares.
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Squares Squared
- Thread starter BigGlenntheHeavy
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takariendire
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- Jun 9, 2010
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Hello, "BigGlenntheHeavy!
This requires some Olympic-level gymnastics . . .
\(\displaystyle \begin{array}{ccccc}\text{We have:} &a^2c^2 + a^2d^2 + b^2c^2 + b^2d^2 \\ \text{Regroup terms:} & (a^2c^2 + b^2d^2) + (a^2d^2 + b^2c^2) \\ \text{Add and subtract }2abcd\!:} & (a^2c^2 + 2abcd + b^2d^2) + (a^2d^2 - 2abcd + b^2c^2) \\ \text{Factor:} & (ac + bd)^2 + (ad - bc)^2 \end{array}\)
\(\displaystyle \begin{array}{ccccc}\text{We have:} &a^2c^2 + a^2d^2 + b^2c^2 + b^2d^2 \\ \text{Regroup terms:} & (a^2c^2 + b^2d^2) + (a^2d^2 + b^2c^2) \\ \text{Subtract and add }2abcd\!:} & (a^2c^2 - 2abcd + b^2d^2) + (a^2d^2 + 2abcd + b^2c^2) \\ \text{Factor:} & (ac - bd)^2 + (ad + bc)^2 \end{array}\)
\(\displaystyle \text{Therefore, there are }two\text{ solutions:}\)
.\(\displaystyle (a^2+b^2)(c^2+d^2) \;=\;\begin{array}{cc} (ac+bd)^2 + (ad-bc)^2 \\ \\[-3mm] (ac-bd)^2 + (ad+bc)^2 \end{array}\)
This requires some Olympic-level gymnastics . . .
\(\displaystyle \text{Show that the product }\,(a^2+b^2)(c^2+d^2)\,\text{ can be written as the sum of two squares.}\)
\(\displaystyle \begin{array}{ccccc}\text{We have:} &a^2c^2 + a^2d^2 + b^2c^2 + b^2d^2 \\ \text{Regroup terms:} & (a^2c^2 + b^2d^2) + (a^2d^2 + b^2c^2) \\ \text{Add and subtract }2abcd\!:} & (a^2c^2 + 2abcd + b^2d^2) + (a^2d^2 - 2abcd + b^2c^2) \\ \text{Factor:} & (ac + bd)^2 + (ad - bc)^2 \end{array}\)
\(\displaystyle \begin{array}{ccccc}\text{We have:} &a^2c^2 + a^2d^2 + b^2c^2 + b^2d^2 \\ \text{Regroup terms:} & (a^2c^2 + b^2d^2) + (a^2d^2 + b^2c^2) \\ \text{Subtract and add }2abcd\!:} & (a^2c^2 - 2abcd + b^2d^2) + (a^2d^2 + 2abcd + b^2c^2) \\ \text{Factor:} & (ac - bd)^2 + (ad + bc)^2 \end{array}\)
\(\displaystyle \text{Therefore, there are }two\text{ solutions:}\)
.\(\displaystyle (a^2+b^2)(c^2+d^2) \;=\;\begin{array}{cc} (ac+bd)^2 + (ad-bc)^2 \\ \\[-3mm] (ac-bd)^2 + (ad+bc)^2 \end{array}\)