From the quadratic formula, we know that the numbers and are the roots of the quadratic equation where if and only if
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
For the sum we have,
Now, for the product, we have
In the next section, we prove that if
then, and are the roots of .
To do this, we multiply both sides of the preceding equations by giving us
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.
Senk, Sharon L., et al. “University of Chicago School Mathematics Project Advanced Algebra.” Glenview, IL: Scott, Foresman and Company (1990).