Question

If $\alpha ,\beta$ are the roots of $a{x}^2+bx+c=0$, the correct statements about the equation a${(x-2)}^2+b(x-2)+c=0$ is

• A. $2\alpha ,2\beta$ are the roots
• B. $\alpha -2,\beta -2$ are the roots
• C. $\alpha +2,\beta +2$ are the roots
• D. $\frac{\alpha }{2},\frac{\beta }{2}$ are the roots

Answer 1

The correct option is C

$\alpha +2,\beta +2$ are the roots

$\alpha ,\beta$ are the roots of a$x^2+bx+c=0$

$\Rightarrow a{\alpha }^2+b\alpha +c=0$ and $a{\beta }^2+b\beta +c=0$ clearly, if we substitute $\alpha +2$

and $\beta +2$ in $a{(x-2)}^2+b(x-2)+c$, we get

$a(\alpha +2-2{)}^2+b(\alpha +2-2)+c=a{\alpha }^2+b\alpha +c=0$

$\Rightarrow \alpha +2$ and $\beta +2$ are the roots.