Question

If both roots of the quadratic equation $a{x}^2+bx+c=0$ are -ve, then prove that $a,b,c$ are of the same sign.

Answer 1

Let $\alpha$ and $\beta$ be the roots of the equation $a{x}^2+bx+c=0$.

Given that $\alpha$ and $\beta$ are negative.

Now,

sum of roots $=-\frac{b}{a}$

$\alpha +\beta =-\frac{b}{a}$

Since $\alpha$ and $\beta$ are negative, their sum will be negative.

For $-\frac{b}{a}$ to be negative, $a$ and $b$ should be of same sign.

Now

Product of roots $=\frac{c}{a}$

Again, since $\alpha$ and $\beta$ are negative, their product will be positive.

Thus, for $\frac{c}{a}$ to be positive, $a$ and $c$ should be of same sign.

Hence, we can say that if both roots of the quadratic equation $a{x}^2+bx+c=0$ are negative, then $a,b,c$ will be of the same sign.

Hence proved.