Sum of Roots of a Quadratic equation

Formula

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

Introduction

In mathematics, $ax^2+bx+c = 0$ is a standard form of a quadratic equation. The highest exponent of variable in a quadratic equation is $2$. So, it always has two roots and they are usually denoted by two Greek alphabets alpha and beta.

The two roots of quadratic equation are written as follows.

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

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

The addition of roots of a quadratic equation is written as $\alpha+\beta$ and it is equal to the negative of the quotient of $b$ divided by $a$.

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

Now, let’s learn how to derive the sum of roots of a quadratic equation.

Proof

Add the two roots of a quadratic equation to find their sum mathematically.

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

The terms on the right-hand side of the equation are like fractions because their denominators are same. So, the fractions can be directly added by the addition rule of like fractions.

$\,\,=\,\,$ $\dfrac{-b+\sqrt{b^2\,-\,4ac}+\big(-b-\sqrt{b^2\,-\,4ac}\big)}{2a}$

Now, let’s simplify this fraction to find the sum of roots of a quadratic equation.

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

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

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

$\,\,=\,\,$ $\dfrac{-2b}{2a}$

$\,\,=\,\,$ $-\dfrac{2b}{2a}$

$\,\,=\,\,$ $-\dfrac{2 \times b}{2 \times a}$

$\,\,=\,\,$ $-\dfrac{\cancel{2} \times b}{\cancel{2} \times a}$

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

Now, let’s learn how to use the sum of roots of quadratic equation as a formula in mathematics.

Example

$4x^2+5x+6 = 0$

Compare this quadratic equation with standard form quadratic equation, then $a = 4$, $b = 5$ and $c = 6$. Now, let’s substitute them in the sum of roots formula.

$\,\,\,\therefore\,\,\,\,\,\,$ $\alpha+\beta$ $\,=\,$ $-\dfrac{b}{a}$ $\,=\,$ $-\dfrac{5}{4}$

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