Question

Which of the following quadratic expressions can be expressed as a product of real linear factors?

A. ${x^2} - 2x + 3$
B. $3{x^2} - \sqrt 2 x - \sqrt 3 $
C. $\sqrt 2 {x^2} - \sqrt 5 x + 3$
D. None of these

Answer 1

Hint- In order to deal with this question we will use the concept as if discriminant of any quadratic equation is greater than or equal to zero then that equation can be expressed as product of real factors so according to it we will check discriminant of each option and find the correct option.

Complete step-by-step answer:
As we know that quadratic expression can be expressed as a product of real linear factors if their discriminant is greater than or equal to zero.
If quadratic equation is given as $a{x^2} + bx + c = 0$ then discriminant can be expressed as
$D = \sqrt {{b^2} - 4ac} $
Now step by step we will check each option
By considering option A
Quadratic equation is ${x^2} - 2x + 3$
After comparing with general equation we get the values as $a = 1,b = - 2$ and $c = 3$
By putting above values in formula of discriminant we get
$
  D = {( - 2)^2} - 4(1)(3) \\
   = 4 - 12 \\
   = - 8 \\
$
Here value of discriminant is less than zero
Thus according to property it can not be expressed as a product of linear factors.
Now by taking option B
Quadratic equation is $3{x^2} - \sqrt 2 x - \sqrt 3 $
After comparing with general equation we get the values as $a = 3,b = - \sqrt 2 $ and $c = - \sqrt 3 $
By putting above values in formula of discriminant we get
\[
   = {( - \sqrt 2 )^2} - 4(3)( - \sqrt 3 ) \\
   = 2 + 12\sqrt 3 \\
\]
Here value of discriminant is greater than zero
Thus according to property it can be expressed as a product of linear factors.
Similarly by taking option C
Quadratic equation is $\sqrt 2 {x^2} - \sqrt 5 x + 3$
After comparing with general equation we get the values as \[a = \sqrt 2 ,b = - \sqrt 5 \] and \[c = 3\]
By putting above values in formula of discriminant we get
\[
   = {( - \sqrt 5 )^2} - 4(\sqrt 2 )(3) \\
   = 5 - 12\sqrt 2 \\
\]
Here value of discriminant is less than zero
Thus according to property it can not be expressed as a product of linear factors.
Hence the correct answer is option B.

Note- For expressing a product of linear factors $D$ should be greater than $0$. If it is less than $0$then we cannot represent it as a product of linear factors . We can determine it by the formula $D = \sqrt {{b^2} - 4ac} $ where $a,b$ and $c$ are coefficients of the quadratic equation.