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