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:

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

\displaystyle\boxed{\alpha\beta=\frac{c}{a}}

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}

Hence,

\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!

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Author: mathtuition88

Math and Education Blog

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