An incompatible sum and pair-product system

Let $a,b,c$ be real numbers and $n,k$ be positive real numbers. Suppose that we have

$a+b+c=n$ $ab+bc+ca=k$

Prove that there are no real $a,b,c$ that satisfy the equations if $k\geq\frac{n^{2}}{2}$ .

Source: InternLM Lean-Workbook, record lean_workbook_5769 (Apache-2.0). The complete source proposition and explicit variable declarations are preserved.