Question

Let $a>0,b>0$ and $c>0$. Then the roots of the equations $a{x}^2+bx+c=0$

• A. are real and negative
• B. have negative real parts
• C. both (a) and (b)
• D. none of these

Answer 1

The correct option is B

have negative real parts

Determine the roots of the equation $a{x}^2+bx+c=0$

Given, $a,b>0$ and $c>0$ and $a{x}^2+bx+c=0$

Therefore the roots of the given equation are:

$x=-\frac{b}{2a}\pm \frac{\sqrt{b^2-4ac}}{2a}$

Case 1: When $b^2-4ac>0$

The roots of the given quadratic equation will be:

$x=-\frac{b}{2a}+\frac{\sqrt{b^2-4ac}}{2a}$ and $x=-\frac{b}{2a}-\frac{\sqrt{b^2-4ac}}{2a}$

Here, both the roots are negative.

Case 2: When $b^2-4ac=0$

The roots of the given equation will be:

$x=-\frac{b}{2a}$

Both the roots will be negative and will be equal.

Case 3: When $b^2-4ac<0$

The roots of the quadratic equation will be:

$x=-\frac{b}{2a}\pm \iota \frac{\sqrt{4ac-b^2}}{2a}$

Here also the roots have a negative real part.

So from the above explanation, we can say that the roots of a given quadratic equation have a negative real part

Hence, option B is the correct answer.