Question

If $\alpha$ and $\beta (\alpha <\beta )$ are the roots of the equation $x^2+bx+c=0$, where $c<0<b$, then

• A. $\alpha <0<\beta <|\alpha |$
• B. $\alpha <0<|\alpha |<\beta$
• C. $\alpha <\beta <0$
• D. $0<\alpha <\beta$

Answer 1

The correct option is A $\alpha <0<\beta <|\alpha |$

Given: $\alpha$ and $\beta (\alpha <\beta )$ are the roots of the equation $x^2+bx+c=0$, where $c<0<b$

Here $D=b^2-4c>0$ because $c<0<b$.
So, roots are real and unequal.
Now, $\alpha +\beta =-b<0$ and $\alpha \beta =c<0$
Therefore, One root is positive and the other negative, the negative root being numerically bigger.
As $\alpha <\beta ,\alpha$ is the negative root while $\beta$ is the positive root.
So,$|\alpha |>\beta$ and $\alpha <0<\beta$.