Question

If $a,b,c$ are all positive and in $H.P.$, then the roots of $a{x}^2+2bx+c=0$ are

• A. real
• B. imaginary
• C. rational
• D. equal

Answer 1

The correct option is A real

$a,b,c$ are $HP$

$b=\frac{2ac}{a+c}$

${ax}^2+2bx+c=0$

$\Delta ={\left(2b\right)}^2-4ac$

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

$=4[\frac{4a^2{c}^2}{{(a+c)}^2}-ac]$ is always positive

$\therefore$ , roots are real