First slide

Theory of equations

Question

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

Moderate

A

are real and positive
B

real and negative
C

have negative real part
D

have positive real part

Solution

Roots of $a x^{2} + b c + c = 0$ are given by
$x = - \frac{b}{2 a} \pm \frac{\sqrt{b^{2} - 4 a c}}{2 a}$
We have,
$a, b > 0 \Rightarrow - b / 2 a < 0$
If $b^{2} - 4 a c < 0$ then roots are imaginary of which real part is negative.
If $b^{2} - 4 a c > 0$ then roots are real and negative since $\sqrt{b^{2} - 4 a c} < b$ .
Hence, in either case, both the roots have negative real part.

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