Question

Which of the following quadratic equations will be having the sum of the roots as 3 and the product of the roots as 2?
$
  {\text{A}}{\text{. }}{x^2} - 3x + 2 = 0 \\
  {\text{B}}{\text{. }}{x^2} + 3x + 2 = 0 \\
  {\text{C}}{\text{. 3}}{x^2} + 6x + 10 = 0 \\
  {\text{D}}{\text{. }}{x^2} + 3x - 2 = 0 \\
 $

Answer 1

Hint- Here, we will simply proceed by using the concept that any general quadratic equation in terms of its sum of the roots and its product of the roots is given by ${x^2}$- (Sum of roots) $x$ + (Product of roots) = 0.

Complete step-by-step solution -
Given, Sum of the roots = 3
Product of the roots = 2
For any general quadratic equation $a{x^2} + bx + c = 0$,
Sum of its roots = $ - \dfrac{b}{a}$
Product of its roots = $\dfrac{c}{a}$
As we know that any general quadratic equation can be written in terms of its sum of roots and product of roots as under
${x^2}$- (Sum of roots)x + (Product of roots) = 0
By substituting the values of the sum of the roots and the product of the roots (given in the problem) in the above equation, we get
$ \Rightarrow {x^2} - 3x + 2 = 0$
The above obtained quadratic equation represents the required quadratic equation.
Hence, option A is correct.

Note- In this particular problem, we can also use another method i.e., finding the sum and the product of the roots of all the quadratic equations given in the options. For ${x^2} - 3x + 2 = 0$, the sum of the roots is 3 and the product of the roots is 2. For${x^2} + 3x + 2 = 0$, the sum of the roots is -3 and the product of the roots is 2. For${\text{3}}{x^2} + 6x + 10 = 0$, the sum of the roots is -2 and the product of the roots is $\dfrac{{10}}{3}$. For${x^2} + 3x - 2 = 0$, the sum of the roots is -3 and the product of the roots is -2.