Question

Find the error in the mathematical expression:
$\left( {x - 4} \right)\left( {x - 2} \right) = {x^2} - 8$

Answer 1

Hint: Perform the multiplication on the left hand side of the equation step by step and check whether the resultant expression is matching with the right hand side or not. If both the sides are matching, then there is no error. And if they are not matching, identify what needs to be fixed so that they become equal.

Complete step-by-step answer:
According to the question, the given mathematical expression is:
$ \Rightarrow \left( {x - 4} \right)\left( {x - 2} \right) = {x^2} - 8$
Separating left hand side and right hand side of the above expression, we have:
$
   \Rightarrow LHS = \left( {x - 4} \right)\left( {x - 2} \right) \\
   \Rightarrow RHS = {x^2} - 8
 $
If we solve the left hand side of the expression systematically, we have:
$ \Rightarrow LHS = x\left( {x - 2} \right) - 4\left( {x - 2} \right)$
Multiplying and solving it further, we’ll get:
$
   \Rightarrow LHS = {x^2} - 2x - 4x + 8 \\
   \Rightarrow LHS = {x^2} - 6x + 8
 $
From this we can clearly see that the left hand side is not equal to the right hand side as:
$
   \Rightarrow {x^2} - 6x + 8 \ne {x^2} - 8 \\
   \Rightarrow LHS \ne RHS
 $
Suppose to make the left hand side equal to the right hand side we have to add $k$ to it. So we have:
$ \Rightarrow LHS + k = RHS$
Putting the value of LHS and RHS, we’ll get:
$
   \Rightarrow {x^2} - 6x + 8 + k = {x^2} - 8 \\
   \Rightarrow k = - 8 - 8 + 6x \\
   \Rightarrow k = 6x - 16
 $
Hence, for the above expression to make right, we need to add $6x - 16$ on the left hand side. Or alternatively we can say that we need to subtract $6x - 16$ on the right hand side.

Thus the error in the above expression is that $6x - 16$ is not added on the left hand side of the equation or $6x - 16$ is not subtracted on the right hand side of the equation.

Note: The right hand side of the above expression is ${x^2} - 8$. This can be further written as:
$ \Rightarrow {x^2} - 8 = {x^2} - {\left( {\sqrt 8 } \right)^2}$
Factorising this with the help of $\left( {{a^2} - {b^2}} \right) = \left( {a + b} \right)\left( {a - b} \right)$, we’ll get:
\[ \Rightarrow {x^2} - 8 = \left( {x + \sqrt 8 } \right)\left( {x - \sqrt 8 } \right)\]
Now we can clearly see that:
\[ \Rightarrow \left( {x - 4} \right)\left( {x - 2} \right) \ne \left( {x + \sqrt 8 } \right)\left( {x - \sqrt 8 } \right)\]
Thus if we have to check the validity of the expression, we can use this factorisation method also.