Question

If \[\alpha \] and $\beta $ are the roots of the equation $a{x^2} + bx + c = 0$, $\alpha \beta = 3$, and a, b, c are in AP, then $\alpha + \beta $ is equal to
A.-4
B.1
C.4
D.-2

Answer 1

Hint: As we know, the sum of the roots of an equation $a{x^2} + bx + c$ is $\dfrac{{ - b}}{a}$ and the product of roots is $\dfrac{c}{a}$. And a, b and c are in AP. So, the difference between the two consecutive terms will be the same. So, by using this we will find the value of $\alpha + \beta $.

Complete step-by-step answer:
Given:
$a{x^2} + bx + c = 0$
$\alpha \beta = 3$
$\alpha $and $\beta $are the roots of the equation $a{x^2} + bx + c$. So,
$\alpha + \beta = \dfrac{{ - b}}{a}$-------(1)
$\alpha \beta = \dfrac{c}{a}$
As $\alpha \beta = 3$, so $\dfrac{c}{a} = 3 \Rightarrow c = 3a$-----(2)
a, b and c are in AP. So, the difference between 2 consecutive terms will be the same.
b-a = c-b
Using equation (2), replacing the value of c in the above equation.
\[b - a = 3a - b\]
$b + b = 3a + a$
$2b = 4a$
$b = 2a$------(3)
Now, according to equation (1),
$\alpha + \beta = \dfrac{{ - b}}{a}$
By using equation (3), replacing the value of b in the above equation.
$\alpha + \beta = \dfrac{{ - 2a}}{a}$
$\alpha + \beta = - 2$
So, the correct answer is “Option D”.

Note: Arithmetic Progression (AP) is a sequence of numbers in order in which the difference of any two consecutive numbers is a constant value.
 There is another method also to solve this question. And the other method to solve this question is by finding the values of $\alpha $and $\beta $separately using the direct formula and then adding them. In this we don’t need to calculate the sum and product of the roots of the equation. But it would be a little bit complicated.
And while substituting and simplifying we should not neglect any terms or sign because it will change the whole result.