QUESTION

# If $\alpha$ and $\beta$ , are the roots of the quadratic equation ${x^2} + bx + c = 0$ where c < 0 < b , thenA. 0 < $\alpha$ < $\beta$B. $\alpha$ < 0 < $\beta$ < $\left| \alpha \right|$C. $\alpha$ < $\beta$ < 0D. $\alpha$ < 0 < $\left| \alpha \right|$ < $\beta$

Given equation is ${x^2} + bx + c = 0$
It is given that $\alpha$ and $\beta$ are the roots of the equation.
$\Rightarrow$ Sum of roots = $\alpha + \beta = - \dfrac{\text{coefficient of x}}{\text{coefficient of }x^2} = \dfrac{{ - b}}{1} = - b$
$\Rightarrow$ Product of roots = $\alpha \beta = \dfrac{\text{constant}}{\text{coefficient of }x^2} = \dfrac{c}{1} = c$
$\Rightarrow$ c < 0
$\Rightarrow$ $\alpha $\beta < 0 \Rightarrow One root is negative Let us suppose \alpha is negative \Rightarrow \beta is positive \Rightarrow \alpha < 0 < \beta …… (2) From equation (1) b > 0 \Rightarrow -b < 0 Now, \alpha + \beta = - b \Rightarrow \alpha + \beta < 0 \Rightarrow -\alpha > \beta \Rightarrow $\left| \alpha \right| > \beta$ …… (3) So, from equation (2) and (3) \Rightarrow$\alpha$ < 0 < $\beta$ < $\left| \alpha \right|$