Let p and q be real numbers such that the roots of x^2 + px + q = 0 are real. Prove that the roots of
x^2 + px + q + (x + a)(2x + p) = 0 are real, for any real number a.

Step-by-step explanation:

Known fact: all roots of quadratic equation are real if and only if

discriminant

We know that all roots of are real. We can derive from this condition that

. (<----------- HERE)

Let's do some simplification of second equation:

So we want according to Known fact prove that discriminant of equation

is greater or equal zero.

.

To prove that , we can view D as polynomial of a:

.

Known fact #2: if quadratic polynomial have positive greatest coefficient and it's discriminant then polynomial always positive.

So remains to prove that

lets divide both side by :

divide by 3

transfer terms to the other side:

But we know that from "(<----------- HERE)"!!!

So Q.E.D.