Let a and b be the roots of the equation x^2 - mx + 2 = 0. Suppose that a + 1/b + ab and b + 1/a + ab are the roots of the equation x^2 - px + q = 0. What is q?
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\(x^2 - mx + 2 = 0\)
p q
\(x=-\frac{p}{2}\pm \sqrt{\frac{p^2}{4}-q}\\ a=\frac{m}{2}+ \sqrt{\frac{m^2}{4}-2}\\ b=\frac{m}{2}- \sqrt{\frac{m^2}{4}-2}\)
\({\color{blue}x^2 - px + q =}\{x-(\frac{m}{2}+ \sqrt{\frac{m^2}{4}-2}+\frac{1}{\frac{m}{2}- \sqrt{\frac{m^2}{4}-2}}\\ +(\frac{m}{2}+ \sqrt{\frac{m^2}{4}-2})(\frac{m}{2}- \sqrt{\frac{m^2}{4}-2}))\}\)
\(\times\{x-(\frac{m}{2}- \sqrt{\frac{m^2}{4}-2}+\frac{1}{\frac{m}{2}+ \sqrt{\frac{m^2}{4}-2} }+(\frac{m}{2}+ \sqrt{\frac{m^2}{4}-2})(\frac{m}{2}- \sqrt{\frac{m^2}{4}-2}))\}\)
\(\sqrt{\frac{m^2}{4}-2}=z\)
\({\color{blue}x^2 - px + q =}\{x-(\frac{m}{2}+ z+\frac{1}{\frac{m}{2}- z} +(\frac{m}{2}+ z )(\frac{m}{2}- z ))\}\)
\(\times\{x-(\frac{m}{2}-z +\frac{1}{\frac{m}{2}+ z }+(\frac{m}{2}+ z )\times (\frac{m}{2}-z ))\}\)
\({\color{blue}x^2 - px + q =} (x-(a + 1/b + ab))(x-( b + 1/a + ab))\)
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