Question

If $a,b$ and $c$ are distinct positive real numbers in AP, then the roots of the equation $a{x}^2+2bx+c=0$ are

• A. imaginary
• B. rational and equal
• C. rational and distinct
• D. irrational

Answer 1

The correct option is C

rational and distinct

Explanation for the correct option:

Finding the roots of the equation:

Given that, $a,b,c$ are in A.P, then

$2b=a+c$

Now, the given equation becomes:

$a{x}^2+(a+c)x+c=0$

So, Discriminant $D=b^2–4ac$

$$\begin{gathered} ={(a+c)}^2-4ac \\ =a^2+c^2+2ac-4ac \\ =a^2+c^2-2ac \\ ={\left(a-c\right)}^2 \end{gathered}$$

Now, ${\left(a-c\right)}^2\ge 0$

$\Rightarrow$ $D\ge 0$

$\therefore$The roots of the equation are rational and distinct.

Hence, option ‘C’ is Correct.