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Nature of Roots

If a, b, c ...

Question

If $a, b, c$ are positive, then both roots of the equation $a x^{2} + b x + c = 0$

are real and negative

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are real and positive

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have negative real parts

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have postive real parts

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Solution

The correct option is C have negative real parts
We know that the roots of a quadratic equation $a x^{2} + b x + c = 0$ is:

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

Now, consider the three cases:

Case 1: If $b^{2} - 4 a c > 0$ then

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

Case 2: If $b^{2} - 4 a c = 0$ then

$x = \frac{- b}{2 a}$ , that is both roots are equal and negative.

Case 3: If $b^{2} - 4 a c < 0$ then

$x = \frac{- b}{2 a} \pm i \frac{\sqrt{{4 a c - b}^{2}}}{2 a}$ have both negative real parts.

Hence, both the roots of the quadratic equation $a x^{2} + b x + c = 0$ have negative real parts.

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