Question

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

Answer 1

Hint: Here we will first use the relation between the roots of the quadratic equation and the coefficients of the quadratic equation. Then we will use the inequalities given in the question. We will use all the inequalities given in the question together and we will get the required result.

Complete step by step solution:
It is given that $\alpha $ and $\beta $ are the two roots of the given equation i.e. ${x^2} + bx + c = 0$
Now, we will first use the relation between the roots of the quadratic equation and the coefficients of the quadratic equation.
We know if $a{x^2} + bx + c = 0$ is a quadratic equation, then the sum of the roots of this equation will be equal to the ratio of the coefficient of $x$ to the coefficient of ${x^2}$ and the product of the roots of the equation will be equal to the constant term to the coefficient of ${x^2}$.
So we will use the same concept.
$ \Rightarrow \alpha + \beta = \dfrac{{ - b}}{1} = - b$ ………….. $\left( 1 \right)$
$ \Rightarrow \alpha \times \beta = \dfrac{c}{1} = c$ ………….. $\left( 2 \right)$
As it is given that $b > 0$
So we can write equation 1 as
$\alpha + \beta < 0$
It is also given that $c < 0$.
So we can write equation 2 as
$\alpha \times \beta < 0$
The product of two numbers is less than zero when one of them is negative.
But we have $\alpha < \beta $ ……………. $\left( 3 \right)$
So we can say that $\beta $ is a positive root and $\alpha $ is a negative root.
So we can write $\alpha < 0$ and $\beta > 0$ …………….. $\left( 4 \right)$
On taking modulus on both sides of the inequalities of equation 3, we get
$\left| \alpha \right| > \left| \beta \right|$
As $\beta $ is a positive root, so we can write it as
$\left| \alpha \right| > \beta $ ………….. $\left( 5 \right)$
Now, we will compare equation 4 and equation 5
$\alpha < 0 < \beta < \left| \alpha \right|$.

Therefore, the correct option is option B.

Note: Here we have obtained the relation between the roots of the quadratic equation. Here roots of the quadratic equation are defined as the values which when put in the place of the variables will satisfy the equation. The number of roots of the polynomial is always equal to the degree of that polynomial.