Question

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

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

Answer 1

Answer: (B)

$\alpha <0<\beta <|\alpha |$

Here $D=b^2-4c>0$ because $c<0<b.$

So, roots are real and unequal.

Now,$\alpha +\beta =-b<0\text{and}\alpha \beta =c<0$

Therefore, one root is positive and the other root is negative, the negative root being numerically bigger.

As $\alpha <\beta ,$ so $\alpha$ is the negative root while $\beta$ is the positive root.

So, $|\alpha |>\beta$

and $\alpha <0<\beta <|\alpha |$.