Question

The value of k, if the expression \[{x^2} + kx + 1\;\] is factorisable into two linear factors, is ______ .
A) \[k \leqslant 2\]
B) \[k \geqslant - 2\]
C) Either \[k \leqslant 2\] or \[k \geqslant - 2\]
D) \[ - 2 \leqslant k \leqslant 2\]

Answer 1

Hint:First we will find the discriminant of the given equation and then put it greater than equal to zero to get the desired interval in k lies.
 For any quadratic equation of the form \[a{x^2} + bx + c = 0\;\]
The discriminant is given by:-
\[D = {b^2} - 4ac\]

Complete step-by-step answer:
The given equation is:
\[{x^2} + kx + 1 = 0\]
Comparing it with standard equation \[a{x^2} + bx + c = 0\;\]we get:-
\[
  a = 1 \\
  b = k \\
  c = 1 \\
 \]
 Now applying the formula of discriminant :
 \[D = {b^2} - 4ac\]
We get:
\[
  D = {k^2} - 4\left( 1 \right)\left( 1 \right) \\
  D = {k^2} - 4 \\
 \]
 A quadratic expression can be expressed as a product of two linear factors if and only if discriminant of the equation is greater than or equal to zero.( \[D \geqslant 0\])
Therefore,
\[
   \Rightarrow {k^2} - 4 \geqslant 0 \\
   \Rightarrow \left( {k - 2} \right)\left( {k + 2} \right) \geqslant 0 \\
   \Rightarrow {\text{Either}}\;k \geqslant 2\;{\text{or}}\;k \leqslant - 2 \\
 \]
Hence option C is the correct option

Note:
*In the quadratic equation, if the discriminant is greater than zero then it has real roots.
*In the quadratic equation, if the discriminant is equal to zero then it has equal roots.
*In the quadratic equation, if the discriminant is less than zero then it has imaginary roots.