Sum of roots and Product of roots of Quadratic Equation

Given a quadratic equation ax^2+bx+c=0 with roots \alpha and \beta, we have:



How do we prove this? It is actually due to the quadratic formula!

Recall that the quadratic formula gives the roots of the quadratic equation as: \displaystyle\boxed{x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}}

Now, we can let

\displaystyle \alpha=\frac{-b+\sqrt{b^2-4ac}}{2a}

\displaystyle \beta=\frac{-b-\sqrt{b^2-4ac}}{2a}


\displaystyle \alpha+\beta=\frac{-2b}{2a}=\frac{-b}{a}

\begin{array}{rcl}  \displaystyle    \alpha\beta&=&\frac{-b+\sqrt{b^2-4ac}}{2a}\times\frac{-b-\sqrt{b^2-4ac}}{2a}\\    &=&\frac{b^2-(b^2-4ac)}{4a^2}\\    &=&\frac{4ac}{4a^2}\\    &=&\frac{c}{a}\end{array}

In the above proof, we made use of the identity (A+B)(A-B)=A^2-B^2

The above formulas are also known as Vieta’s formulas (for quadratic). There we have it, this is how we prove the formula for the sum and product of roots!


Author: mathtuition88

Math and Education Blog

2 thoughts on “Sum of roots and Product of roots of Quadratic Equation”

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.

%d bloggers like this: