Question

Let $a>0,b>0,c>0$ then both roots of the equation $a{x}^2+bx+c=0$

• A. are real and negative
• B. have negative real parts
• C. have positive real parts
• D. none of these

Answer 1

The correct option is B have negative real parts

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

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

$Case1:$ When $b^2-4ac>0$

$\Rightarrow$ $x=\frac{-b}{2a}-\frac{\sqrt{b^2-4ac}}{2a}$ and $\frac{-b}{2a}+\frac{\sqrt{b^2-4ac}}{2a}$ both roots are negative.

$Case2:$ Ehen $b^2-4ac=0$

$\Rightarrow$ $x=\frac{-b}{2a}$ i.e., both roots are equal and negative.

$Case3:$ When $b^2-4ac<0$

$\Rightarrow$ $x=\frac{-b}{2a}\pm i\frac{\sqrt{4ac-b^2}}{2a}$ have negative real part.

$\therefore$ From above discission both roots have negative real parts.