Question
Answer
$$\dfrac{(a^2+b^2+c^2+d^2)^2}{1+ac+bd}=\dfrac{a^4+2a^2b^2+2a^2c^2+2a^2d^2+b^4+2b^2c^2+2b^2d^2+c^4+2c^2d^2+d^4}{1+ac+bd}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{(a^2+b^2+c^2+d^2)^2}{1+ac+bd}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{a^4+2a^2b^2+2a^2c^2+2a^2d^2+b^4+2b^2c^2+2b^2d^2+c^4+2c^2d^2+d^4}{1+ac+bd}\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{a^2+b^2+c^2+d^2}\right) $ by each term in $ \left( a^2+b^2+c^2+d^2\right) $. $$ \left( \color{blue}{a^2+b^2+c^2+d^2}\right) \cdot \left( a^2+b^2+c^2+d^2\right) = \\ = a^4+a^2b^2+a^2c^2+a^2d^2+a^2b^2+b^4+b^2c^2+b^2d^2+a^2c^2+b^2c^2+c^4+c^2d^2+a^2d^2+b^2d^2+c^2d^2+d^4 $$ |
| ② | Combine like terms: $$ a^4+ \color{blue}{a^2b^2} + \color{red}{a^2c^2} + \color{green}{a^2d^2} + \color{blue}{a^2b^2} +b^4+ \color{orange}{b^2c^2} + \color{blue}{b^2d^2} + \color{red}{a^2c^2} + \color{orange}{b^2c^2} +c^4+ \color{red}{c^2d^2} + \color{green}{a^2d^2} + \color{blue}{b^2d^2} + \color{red}{c^2d^2} +d^4 = \\ = a^4+ \color{blue}{2a^2b^2} + \color{red}{2a^2c^2} + \color{green}{2a^2d^2} +b^4+ \color{orange}{2b^2c^2} + \color{blue}{2b^2d^2} +c^4+ \color{red}{2c^2d^2} +d^4 $$ |
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