Question

If $a,b,c\in R^{+}$ and $2b=a+c$, then check the nature of roots of equation $a{x}^2+2bx+c=0$.

Answer 1

If $a,b,c\in R^{+}$

and $2b=a+c------\left(1\right)$

equation, $a{x}^2+2bx+c=0$

Nature of roots

$D=B^2-4AC$

$=(2b{)}^2-4ac$

$=4b^2-4ac$

$=(a+c{)}^2-4ac$

$=a^2+c^2+2ax-4ac$

$=a^2+c^2-2ac$

$=(a-c{)}^2>0$

then, real roots

Hence, this is the answer.