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 $ax^{\displaystyle 2}+bx+c = 0$.

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

Latest Math Topics

Latest Math Problems

Email subscription

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.