Quadratic Equation Roots Prove

Question

I have a question in my textbook from chapter of quadratic equations from exercise of sum of roots and product of roots that; Prove that the equation $$ a x^2 + b x + c = 0, \quad a > 0 $$

has

1. both roots positive, iff $b < 0$ and $c > 0$.

2. both roots negative, iff $b > 0$ and $c > 0$

3. one root positive and other negative, iff $c > 0$

And what will be the roots when $b < 0$ and $c < 0$

I am stuck in this question somebody help me please.

Answer 1

As $a>0$, $c>0\iff x_1x_2>0 \iff x_1x_2\enspace$ have the same sign. Note in such a case this sign is also the sign of their sum, $-\dfrac ba$.

Answer 2

HINT:

If $p$ and $q$ are roots of the quadratic equation then,

$p+q=-\frac{b}{a}$

$pq=\frac{c}{a}$

For the $(i)$ part:

$p+q=-\frac{b}{a} >0$ (since $b<0$ and $a>0$)....$(1)$

Also $pq=\frac{c}{a} >0$ (since $c>0$ and $a>0$)($\implies$ either they are both positive or both negative)....$(2)$

Combining $(1)$ and $(2)$, we have that $p$ and $q$ are positive.

Can you now do the other two yourself?