Question

The sum of roots of a quadratic equation $a{x^2} + bx + c = 0$ is _ _ _ _ _.

Answer 1

Hint: Here, we are given a quadratic equation and we have to find the sum of its roots. Firstly, we have to find both roots of a given quadratic equation by the use of the Sridharacharya formula. Then, we can simply add both roots of the equation to get the required result.

Formula used:
The roots of a quadratic equation are given by $\dfrac{{ - b \pm \sqrt D }}{{2a}}$ where $D$ is the discriminant which is equal to ${b^2} - 4ac$. Here, $a$ is the coefficient of ${x^2}$ , $b$ is the coefficient of $x$ and $c$ is the constant term of a quadratic equation.

Complete step-by-step answer:
Here, the given quadratic equation is $a{x^2} + bx + c = 0$.
Now, the coefficient of ${x^2}$ is $a$, the coefficient of \[x\] is $b$ and the constant term is $c$.
So, by applying the above given Sridharacharya formula. We can write the roots of a given quadratic equation as $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$.
Now, the first root is $\dfrac{{ - b + \sqrt {{b^2} - 4ac} }}{{2a}}$ and the second root is $\dfrac{{ - b - \sqrt {{b^2} - 4ac} }}{{2a}}$.
We have to find the sum of roots. So simply adding both the roots we get $\dfrac{{ - b + \sqrt {{b^2} - 4ac} }}{{2a}} + \dfrac{{ - b - \sqrt {{b^2} - 4ac} }}{{2a}}$
The sum of roots is $\dfrac{{ - b + \sqrt {{b^2} - 4ac} - b - \sqrt {{b^2} - 4ac} }}{{2a}} = \dfrac{{ - 2b}}{{2a}} = - \dfrac{b}{a}$.

Thus, the sum of roots of a given quadratic equation is $ - \dfrac{b}{a}$.

Note:
Similarly, we can find the product of roots.
The product of roots is $\left( {\dfrac{{ - b + \sqrt {{b^2} - 4ac} }}{{2a}}} \right) \times \left( {\dfrac{{ - b - \sqrt {{b^2} - 4ac} }}{{2a}}} \right)$.
By using $\left( {a + b} \right)\left( {a - b} \right) = {a^2} - {b^2}$ we can write the products of roots as \[\dfrac{{{{\left( { - b} \right)}^2} - \left( {\sqrt {{b^2} - 4ac} } \right)}}{{4{a^2}}} = \dfrac{{{b^2} - \left( {{b^2} - 4ac} \right)}}{{4{a^2}}} = \dfrac{{4ac}}{{4{a^2}}} = \dfrac{c}{a}\].
Thus, the product of the roots of a quadratic equation is given by $\dfrac{c}{a}$.
If discriminant $D = {b^2} - 4ac$ is equal to zero then both the roots of the quadratic equation is equal.