Thread: Further maths. Cubics. What?
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Old October 3rd, 2008, 09:42 AM
Soroban Soroban is offline
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Hello, djmccabie!

You did some awesomely excellent work!



Here's one more fact we can dig up . . .


We know: .\begin{array}{cccc}a + b + c &=& \text{-}2 & {\color{blue}[1]}\\ \\[-4mm] ab+bc + ac &=& \text{-}\frac{4}{3} & {\color{blue}[2]}\\ \\[-4mm] abc &=& \frac{7}{3} & {\color{blue}[3]} \end{array}


Square [1]: .(a + b + c)^2 \;=\;(\text{-}2)^2 \quad\Rightarrow\quad a^2 + 2ab + 2ac + b^2 + 2bc + c^2\:=\:4


\text{We have: }\;a^2+b^2+c^2 + 2\underbrace{(ab + bc + ac)}_{\text{This is -}\frac{4}{3}} \:=\:4 \quad\Rightarrow\quad a^2 + b^2 + c^2 + 2\left(\text{-}\frac{4}{3}\right) \:=\:4


Therefore: .a^2+b^2+c^2\;=\;\frac{20}{3}
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