Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 6.2 National olympiad Prove it

Lemma 10.3. If the quadratic irrational $\alpha$ is a root of the polynomial $A x^{2}+B x+C=0$ , then the other root of this polynomial is $\alpha^{\prime}$ , the conjugate of $\alpha$ .

Solution

Proof. From the quadratic formula, we see that the two roots of $A x^{2}+B x+C=0$ are
$\frac{-B \pm \sqrt{B^{2}-4 A C}}{2 A}$

If $\alpha$ is one of these roots, then $\alpha^{\prime}$ is the other root, because the sign of $\sqrt{B^{2}-4 A C}$ is reversed to obtain $\alpha^{\prime}$ from $\alpha$ .

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