Proving the existence of polynomial $p(x)$ and $q(x)$ such that $f(x)=1/2[p(x)+q(x)]$

Question

Let $f(x)=x^2+ax+c$ where $a,c$ are real numbers. Prove that there exist quadratic polynomials $p(x)$ and $q(x)$ (with real coefficients) having all roots real such that $f(x)=\dfrac{1}{2}[p(x)+q(x)].$

I can't even think from where to start. Can anyone please provide a hint as for where to start from?

THANKS

Answer 1

Without loss of generality by changing $x$ into $x+a/2$, we can suppose that $a=0$ and write $f(x)=x^2+c$.

Now take $p(x) =x^2+Bx +c$ and $q(x)=x^2 -Bx +c$ with $B^2 \gt 4c$. This ensures that the discriminants of $p,q$ are positive and that $p,q$ have real roots.