# Sum of Roots of a Quadratic equation

The summation of the two roots (or zeros) of a quadratic equation is called the sum of the roots of a quadratic equation.

## Introduction

According to the quadratic formula, the roots of a quadratic equation are written as follows, when the quadratic equation is expressed as $ax^2+bx+c = 0$, and its two zeros are denoted by alpha and beta.

$\alpha$ $\,=\,$ $\dfrac{-b + \sqrt{b^{\displaystyle 2} \,-\, 4ac}}{2a}$

$\beta$ $\,=\,$ $\dfrac{-b \,- \sqrt{b^{\displaystyle 2} \,-\, 4ac}}{2a}$

For evaluating the sum of the zeros of the quadratic equation, add the two roots alpha and beta.

$\alpha + \beta$ $\,=\,$ $\dfrac{-b + \sqrt{b^{\displaystyle 2} \,-\, 4ac}}{2a}$ $+$ $\dfrac{-b \,- \sqrt{b^{\displaystyle 2} \,-\, 4ac}}{2a}$

$\implies$ $\alpha + \beta$ $\,=\,$ $\dfrac{-b + \sqrt{b^{\displaystyle 2} \,-\, 4ac} -b \,- \sqrt{b^{\displaystyle 2} \,-\, 4ac}}{2a}$

$\implies$ $\alpha + \beta$ $\,=\,$ $\dfrac{-b + \cancel{\sqrt{b^{\displaystyle 2} \,-\, 4ac}} -b \,- \cancel{\sqrt{b^{\displaystyle 2} \,-\, 4ac}}}{2a}$

$\implies$ $\alpha + \beta \,=\, \dfrac{-\, b -b}{2a}$

$\implies$ $\alpha + \beta \,=\, \dfrac{-2b}{2a}$

$\implies$ $\alpha + \beta \,=\, -\dfrac{\cancel{2}b}{\cancel{2}a}$

$\,\,\, \therefore \,\,\,\,\,\, \alpha + \beta \,=\, -\dfrac{b}{a}$

Therefore, it is evaluated that the sum of the two roots of the quadratic equation is $-\dfrac{b}{a}$ for the quadratic equation $ax^{\displaystyle 2}+bx+c = 0$.

### Example

Find the summation of roots of the quadratic equation $4x^2 + 5x + 6 = 0$.

Compare the given quadratic equation with $ax^2 + bx + c = 0$. The values of $a = 4$, $b = 5$ and $c = 6$.

The summation of roots of quadratic equation is $–b/a$.

$-\dfrac{b}{a} = -\dfrac{5}{4}$

Therefore, the addition of the roots of the quadratic equation $4x^2 + 5x + 6 = 0$ is $-\dfrac{5}{4}$.

Email subscription
Math Doubts is a free math tutor for helping students to learn mathematics online from basics to advanced scientific level for teachers to improve their teaching skill and for researchers to share their research projects. Know more