Question

If the equation $a{x}^2+2bx+c=0;a,b,c\in R$ has real roots and $m,n$ are real such that $m^2>n>0$, then the equation $a{x}^2+2mbx+nc=0$ has

• A. real roots
• B. real and equal roots
• C. real and distinct roots
• D. imaginary roots

Answer 1

The correct option is C real and distinct roots

$a{x}^2+2bx+c=0\dots \left(1\right)$
Since the roots of equation $\left(1\right)$ are real,
$\therefore 4b^2-4ac\ge 0$
$\Rightarrow 4b^2\ge 4ac\dots \left(2\right)$

Now, the discriminant of the equation
$a{x}^2+2mbx+nc=0$ is
$\Delta =4m^2{b}^2-4anc$

Also, $m^2>n>0\dots \left(3\right)$
From $\left(2\right)$ and $\left(3\right)$,
$m^2\cdot 4b^2>4ac\cdot n$
$\Rightarrow 4m^2{b}^2-4acn>0$
$\Rightarrow \Delta >0$

Hence, roots of $a{x}^2+2mbx+nc=0$ are real and distinct.