Let $k$ be an algebraically closed field. Let $k[x_1,x_2,x_3]$ be a polynomial ring. Show that $x_1x_3-x_2^2$ is irreducible in $k[x_1,x_2,x_3]$.
I viewed the polynomial $x_1x_3-x_2^2$ as a polynomial in $k[x_1x_3][x_2]=k[x_2][x_1x_3]$. Then it is a polynomial of degree $1$ in $k[x_2][x_1x_3]$. But how do I show that this is equivalent to $x_1x_3-x_2$ being irreducible in $k[x_1,x_2,x_3]$