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Sum of Roots of a Quadratic equation

The summation of the two roots of a quadratic equation is called the sum of the roots of a quadratic equation.

Alpha ($\alpha$) and Beta ($\beta$) are two roots of a quadratic equation $ax^2+bx+c = 0$ as per the quadratic formula.

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

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

Add the two roots of the quadratic equation to obtain the summation of them.

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

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

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

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

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

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

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

Therefore, 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 $-\dfrac{5}{4}$.