Question

If the difference between the roots of the equation \[{x^2} + ax + 1 = 0\] is less than \[\sqrt 5 \], then set of possible values of \[a\] is
A. \[(-3, 3)\]
B. \[( - 3,\infty )\]
C. \[(3,\infty )\]
D. \[( - \infty , - 3)\]

Answer 1

Hint: Here the question is in the form of a quadratic equation and they have given that the difference between the roots of the equation is \[\sqrt 5 \]. So, by using the formula \[|\alpha - \beta | = \sqrt {{{(\alpha + \beta )}^2} - 4\alpha \beta } \] we are going to find the value of \[a\].

Formula Used:
\[|\alpha - \beta | = \sqrt {{{(\alpha + \beta )}^2} - 4\alpha \beta } \]
In quadratic equation \[{ax^2} + bx + c = 0\] if \[\alpha \] and \[\beta \] are the two roots, then
 \[\alpha + \beta\] = \[ \dfrac{-b}{a} \]
\[\alpha \beta\] = \[\dfrac{c}{a}\]

Complete step by step Solution:
Given quadratic equation : \[{x^2} + ax + 1 = 0\]

Let us consider \[\alpha \] and \[\beta \] are the two roots of the given quadratic equation.

The sum of the two roots of the equation = \[\alpha + \beta = -a\]
The product of the two roots of the equation = \[\alpha \beta = 1\]

On considering the formula
\[|\alpha - \beta | = \sqrt {{{(\alpha + \beta )}^2} - 4\alpha \beta } \]
Substituting the known values to the above formula, we have
\[\Rightarrow\] \[|\alpha - \beta | = \sqrt {{{(-a)}^2} - 4 \times 1} \]
\[\Rightarrow\] \[|\alpha - \beta | = \sqrt {{a^2} - 4} \]

\[\sqrt {{a^2} - 4} < \sqrt 5 \] … [Given]

Squaring on both sides
\[\Rightarrow\] \[{a^2} - 4 = 5\]
\[\Rightarrow\] \[{a^2} = 9\]
\[\Rightarrow\] \[ a = \pm 3\]
Therefore the possible values of \[a\] is \[ (-3 , 3)\]


Hence the option A is correct.

Note: The general quadratic equation is in the form of \[a{x^2} + bx + c = 0\]. The formula for finding the roots of the equation is \[x = \dfrac{{b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]. The nature of the root will depend on the discriminant .