Question

If $\alpha ,\beta$ are the roots of the equation $a{x}^2+bx+c=0$ then the quadratic equation whose roots are $\alpha +\beta ,\alpha \beta$ is:

• A. $a^2{x}^2+a(b-c)x+bc=0$
• B. $a^2{x}^2+a(b-c)x-bc=0$
• C. $a{x}^2+(b+c)x+bc=0$
• D. $a{x}^2-(b+c)x-bc=0$

Answer 1

The correct option is B $a^2{x}^2+a(b-c)x-bc=0$

From the first equation, we can conclude that
$\alpha +\beta =-\frac{b}{a}$ ...(i)

$\alpha \beta =\frac{c}{a}$ ...(ii)

Therefore, the new equation will be

$x^2-(\alpha +\beta +\alpha \beta )x+(\alpha +\beta )\left(\alpha \beta \right)=0$

Substituting the values from (i) and (ii), we get

$x^2-\left(\frac{-b+c}{a}\right)x+\left(\frac{-bc}{a^2}\right)=0$

$a^2{x}^2+a(b-c)x-bc=0$

Hence, the answer is
$a^2{x}^2+a(b-c)x-bc=0$