Question

If the roots of the quadratic equation $a{x}^2+bx+c=0$ are opposite in sign and negative root has greater magnitude, then which of the following is/are correct?

• A. $a>0,b>0\text{and}c<0$
• B. $a<0,b<0\text{and}c>0$
• C. $b>0,c>0\text{and}a<0$
• D. $b<0,c<0\text{and}a>0$

Answer 1

Answer: (A), (B)

$a<0,b<0\text{and}c>0$

Given equation $a{x}^2+bx+c=0$
Let the roots be $\alpha ,\beta$
Assuming the greater root to be $\beta$
So, $\beta <0\Rightarrow \alpha >0$
Now,
$\alpha +\beta <0\Rightarrow -\frac{b}{a}<0\Rightarrow \frac{b}{a}>0$
So, $a,b$ are of same sign.
$\alpha \times \beta <0\Rightarrow \frac{c}{a}<0$
So, $a,c$ are of opposite sign.

Therefore,
$a>0,b>0\text{and}c<0$
$a<0,b<0\text{and}c>0$