How to form a quadratic equation with real coefficients if $x_1=4-7i$?

Question

Why is the quadratic equation $x^2-8x+65=0$? I tried to find $p$ and $q$ to form the equation but i need $x_2$ because: $$p=-(x1+x2)$$ $$q=x_1*x_2$$ so $x2=$?

Answer 1

The only way that a polynomial with real coefficients can have a non-real root is if the conjugate is also a root.

So if $4-7i$ is a root, then so is $4+7i$.

Answer 2

If $p(a+bi)=0$ than $p(a-bi)=0$ or differently if $z$ is a root than so is $\overline{z}$