
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
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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.
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.
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