Question
Let $$a > 0, b > 0$$ and $$c > 0.$$ Then both the roots of the equation $$ax^2 + bx + c = 0$$
A.
are real and negative
B.
have negative real parts
C.
are rational numbers
D.
None of these
Answer :
have negative real parts
Solution :
Let, $$a > 0, b > 0, c > 0$$
Given equation $$ax^2 + bx + c = 0$$
we know that $$D = b^2 - 4ac$$ and $$x = \frac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$$
$$\eqalign{
& {\text{Let, }}{b^2} - 4ac > 0,b > 0 \cr
& {\text{If, }}a > 0,c > 0{\text{ then }}{b^2} - 4ac < {b^2} \cr} $$
⇒ Roots are negative
$${\text{Let, }}{b^2} - 4ac < 0,{\text{ then }}x = \frac{{ - b \pm i\sqrt {4ac - {b^2}} }}{{2a}}$$
⇒ roots are imaginary and have negative real part. $$\left( {\because b > 0} \right).$$