Question

lf $\alpha ,\beta$ 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 $a^2{x}^2+a(b-c)x-bc=0$

Given quadratic equation: $a{x}^2+bx+c=0$

$\alpha +\beta =\frac{-b}{a}$ and $\alpha \beta =\frac{c}{a}$

Now,

$\alpha +\beta +\alpha \beta =\frac{-b}{a}+\frac{c}{a}=\frac{c-b}{a}$ and

$(\alpha +\beta )\left(\alpha \beta \right)=\frac{-b}{a}\times \frac{c}{a}=\frac{-bc}{a^2}$

Hence, the quadratic equation whose roots are $\alpha +\beta ,\alpha \beta$ is

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

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

Hence, B is the correct option.