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Let  p and q be real numbers such that the roots of

 

x^2+px+q=0


are real. Prove that the roots of

 

x^2+px+q+(x+a)(2x+p)=0


are real, for any real number a

 Jul 24, 2020
 #1
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By the quadratic formula, the roots are equal to

−p±p2−4q−−−−−−√2−p±p2−4q2

So, since the square root and pp are both integers (since otherwise the roots are not rational). But these two integers are either both even or both odd (why?), and therefore their sum or difference is always even - hence your claim.

 Jul 24, 2020

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