One root of $x^2 + px + q = 0$, for $p, q$ real, is $x = 2 + 3i$. How do you find $p$ and $q$?

solution-verification complex-numbers quadratics

Note that if $\alpha$ and $\beta$ are the roots, then we have

$(x-\alpha)(x-\beta)=0$

$$x^2-(\alpha+\beta) x+\alpha \beta = 0$$

Since your equation is monic,

$$p=-(2+3i+2-3i)=-4$$

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