Question

What is the reciprocal sum of the roots for the equation $2{x^2} + 4x = 0$?
A. $ - \dfrac{1}{2}$
B. $2$
C. $ - 1$
D. $\dfrac{1}{2}$

Answer 1

Hint: Here, in the given question, we need to find the reciprocal sum of the roots for the equation $2{x^2} + 4x = 0$. At first we will find the roots of the given equation.After this, we will find the sum of the roots of the given equation. At the end, we will reciprocate the sum to get our required answer.

Complete step by step answer:
Given equation, $2{x^2} + 4x = 0$. Now we will take $2x$ as a common factor. Therefore, we get
$ \Rightarrow 2x\left( {x + 2} \right) = 0$
$ \Rightarrow 2x = 0$ and $x + 2 = 0$
From here, we get
$ \Rightarrow x = 0$ and $x = - 2$.
These are the roots of the given equation. As here, we need to find the reciprocal sum of the roots for the equation $2{x^2} + 4x = 0$, let us find the sum of the roots of the given equation and to get our required answer we will reciprocate the sum.
Sum of the roots = $0 + \left( { - 2} \right) = - 2$
Reciprocal of $ - 2$ is $ - \dfrac{1}{2}$.
Thus, our required answer is $ - \dfrac{1}{2}$.

Therefore, the correct option is A.

Note: As here, we were given a quadratic equation and we found the roots by simply taking $2x$ as a common factor as one term was zero. If we were given any other term instead of zero, we would have used any other method to find the roots. A quadratic equation of the form $a{x^2} + bx + c = 0$ where $a \ne 0$ can be solved by many methods. Such as, completing the square method, factorization method, by using quadratic formula, etc. Doesn’t matter with which method we solve any quadratic equation because the roots of the quadratic equation will remain the same. The key to solve this type of question is to find the correct roots.