Prove that if one solution for a quadratic equation of the form x 2 + bx + c = 0 is rational (where b and c are rational), then the other solution is also rational. (use the fact that if the solutions of the equation are r and s, then x 2 + bx + c = (x − r)(x − s).)

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05-08-2017
Mathematics

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Prove that if one solution for a quadratic equation of the form x 2 + bx + c = 0 is rational (where b and c are rational), then the other solution is also rational. (use the fact that if the solutions of the equation are r and s, then x 2 + bx + c = (x − r)(x − s).)

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PhosphoricBronze PhosphoricBronze

05-08-2017

[tex]x^2+bx+c = (x-r)(x-s)\\ x^2+bx+c = x^2-(r+s)x+rs\\\\ b = -(r+s)\\ c = rs[/tex]

Say r is rational. Suppose for a second, that s is not. Then, r+s is irrational. But this contradicts the fact that b is rational.

So, if one root is rational, then the other root is also rational

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