Question

If $\alpha $ and $\beta $ are sum and product of roots of the given equation respectively then $( - \alpha \beta )$ is
A.Always a prime number
B.Always an odd integer
C.Always an irrational number
D.Dependent on value of a

Answer 1

Hint: Here we will standard quadratic equation and the sum of the roots and product of the roots formula. Follow step by step approach first sum of the roots, product of roots and then place the values in the given conditions and simplify accordingly.

Complete step-by-step answer:
Let us consider the general form of the quadratic equation.
$a{x^2} + bx + c = 0$
Let us suppose that the roots of the above equation are “p” and “q”.
Now, given that sum of roots is $ = \alpha $
Place the sum of roots from the above standard equation.
$ \Rightarrow \dfrac{{ - b}}{a} = \alpha $
Rewrite the above equation –
$ \Rightarrow \alpha = \dfrac{{ - b}}{a}{\text{ }}.....{\text{ (i)}}$
Now, similarly given that product of roots is $ = \beta $
Place the product of the roots from the above standard equation.
$ \Rightarrow \dfrac{c}{a} = \beta $
Rewrite the above equation –
$ \Rightarrow \beta = \dfrac{c}{a}{\text{ }}.....{\text{ (ii)}}$
Now, place the values from the equation (i) and (ii) in the given condition -
$( - \alpha \beta ) = - \left( {\dfrac{{ - b}}{a} \times \dfrac{c}{a}} \right)$
Product of minus and minus gives us the positive value and Simplify the above equation –
$( - \alpha \beta ) = \left( {\dfrac{{bc}}{{{a^2}}}} \right)$
Thus, value depends on all the a, b and c terms.
Hence, from the given multiple choices – the option D is the correct answer.

Note: Always remember the standard equation and formula properly. Follow the given data and conditions carefully.
A quadratic equation is an equation of second degree, it means at least one of the terms is squared. Standard equation is $a{x^2} + bx + c = 0$ where a,b,and c are constant and “a” can never be zero and “x” is unknown.