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Quadratic Equation & Vieta Theorem

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Kindly elaborate on the concept of squared variables in a system of equations by the sample that was given to me. See attached file for full problem description.

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Solution Summary

The concepts of square variables in a system of equations are found. Square variables are determined.

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I understand your question as a request on how to deal with this system of equations correctly and with understanding.

The second equation, xy = 6 has no squares so that x and y are uniquely related.
We can say y = 6/x and substitute it into the 1st equation:

x^2 +36/x^2 = 25

Now we multiply this by x^2, move everything to the left hand side and get

x^4 - 25 x^2 + 36 = 0

Formally, this is an equation of the 4th order, but in fact it can be reduced to 2nd order by substitution
z = ...

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