Question

If \[\alpha ,\beta \] be the roots of the quadratic equation \[a{x^2} + bx + c = 0\] and k be a real number, then the condition so that \[\alpha < k < \beta \] is given by
A. \[ac > 0\]
B. \[a{k^2} + bk + c = 0\]
C. \[ac < 0\]
D. \[{a^2}{k^2} + abk + ac < 0\]

Answer 1

Hint: In our case, the roots are said to be \[\alpha ,\beta \] of the quadratic equation \[a{x^2} + bx + c = 0\] and k be the real number and are to determine the condition of \[\alpha < k < \beta \] for that we have to first consider the equation as \[P(x) = a{x^2} + bx + c\] and we should consider the roots as \[P(\alpha ) = 0\] OR \[P(\beta ) = 0\]and should apply all the necessary condition \[\alpha < k < \beta \] to get the desired answer.

Formula Used:
Conditions used:
For,
\[\alpha < k < \beta \] AND\[P(k) < 0\]
\[a > 0\;\;\]

Complete Step-By-Step Solution:
Here, in this question it is provided the statement that,
The terms \[{\rm{a}},{\rm{b}}\] be the roots of the quadratic equation \[a{x^2} + bx + c = 0\] and k be a real number
And we should determine the condition that
\[\alpha < k < \beta \]
Now, let us consider the equation to be
\[P(x) = a{x^2} + bx + c\]
Then, the roots can be written as
\[P(\alpha ) = 0\]
OR
\[P(\beta ) = 0\]
Now, we have to apply the given condition, we have
When,
\[a > 0\;\;\]
For,
\[\alpha < k < \beta \]
And
\[P(k) < 0\]
On applying the condition, we get
\[a{k^2} + bk + c\]
Thus, the above obtained equation implies
\[(a{k^2} + bk + c)(a) < 0\]
Therefore, If \[\alpha ,\beta \] be the roots of the quadratic equation \[a{x^2} + bx + c = 0\] and k be a real number, then the condition so that \[\alpha < k < \beta \] is given by \[{a^2}{k^2} + abk + ac < 0\]
Hence, the option D is correct

Note: The best method for determining the condition for the specified result whenever we encounter problems of this nature is to apply the concept. Students often get confused in these types of problem because, it includes more conditions to be applied in order to bring out the results. In such cases, we have to understand the concept and then we should apply the necessary conditions of \[\alpha < k < \beta \] it holds. So, on applying the conditions correctly will give the desired result.