What is the sum of the non-integral solutions of the equation

\( \left| \begin{array}{ccc} x & 3 & 4 \\ 5 & x & 5 \\ 4 & 2 & x \end{array} \right|=0\hspace{2mm}?\)

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TopNewestReduce to a \(2 \times 2\) matrix: \(\left[ \begin{array}{cc} 10-4x & x^2-10 \\ 12-2x & 8-3x \end{array} \right]= 0\)

Then to a \(1 \times 1\) matrix: \((8-3x)(10-4x)-(12-2x)(x^2-10) = 0\)

Which then becomes \(x^3 - 41x + 100 = 0\)

Then after trial and error, x = 4 is one of the non-zero solutions

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Please see the edit now..And please could you give a solution other than hit and trial??Like using synthetic division??

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This solution was created using the Carroll-Dodson Condensation Method. Synthetic division in this case has no meaningful purpose. However, I will see what I can do as far as posting a better solution.

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