From the quadratic formula, we know that the numbers and are the roots of the quadratic equation where if and only if

and

.

In this proof, we are going to show that they are indeed the root of the quadratic equation. We call this the Sum and Product of Roots Theorem.

The equations above come from the fact that the roots of the quadratic equation are

and .

For the sum we have,

.

Now, for the product, we have

.

In the next section, we prove that if

and

then, and are the roots of .

To do this, we multiply both sides of the preceding equations by giving us

(1) and

(2) .

From (1) , implies that .

Substituting the value of in equation (2), we have

.

which is equivalent to

.

Subtracting from both sides and rearranging the terms, we have

.

Multiplying both sides by , we have

This shows that is a root of .

Proving that is also a root is left as an exercise.

**Reference**

Senk, Sharon L., et al. “University of Chicago School Mathematics Project Advanced Algebra.” *Glenview, IL: Scott, Foresman and Company* (1990).