Question

Evaluate the expression \[{\left( {x - 2y - z} \right)^2}\].

Answer 1

Hint: To solve this kind of expression, first of all we need to expand the given expression. To expand the expression we apply the formula \[{\left( {a - b - c} \right)^2} = {a^2} + {b^2} + {c^2} - 2ab + 2bc - 2ac\]. After expanding the expression just solve further till we reach an expression where the further simplifications cannot be done.
Formula used: To solve and simplify the expressions of the form \[\left( {x - 2y - z} \right)\], we have used the following formula. \[{\left( {a - b - c} \right)^2} = {a^2} + {b^2} + {c^2} - 2ab + 2bc - 2ac\]

Complete step-by-step solution:
Now here, since we are given to evaluate an expression which is given in form of whole square, hence first of all we expand the above given expression by applying the formula \[{\left( {a - b - c} \right)^2} = {a^2} + {b^2} + {c^2} - 2ab + 2bc - 2ac\]
Now, substituting \[x\] as \[a\], \[y\] as \[b\] and \[z\] as \[c\] in the above written formula
\[{\left( {x - 2y - z} \right)^2} = {x^2} + 4{y^2} + {z^2} - 4xy + 4yz - 2xz\]
Now, since we cannot further simplify the above we say that the value of the given expression is \[{x^2} + 4{y^2} + {z^2} - 4xy + 4yz - 2xz\].

Note: This expression can also be solved if we do not remember the above mentioned formula. We can also solve it by multiplying \[\left( {x - 2y - z} \right)\] with the expression \[\left( {x - 2y - z} \right)\] itself. To do so first of all we expand it by multiplying each term of \[\left( {x - 2y - z} \right)\] by another term of \[\left( {x - 2y - z} \right)\]. But this method should not be preferred over using formula method because this is a lengthy process and might lead to error in calculation. Remembering formulas can be very useful in further studies when calculations and simplifications become much tougher and complicated.
Additional information: Similarly, to evaluate the expressions of other forms of the same type, we should also remember the following formula.
\[{\left( {x + y + z} \right)^2} = {x^2} + {y^2} + {z^2} + 2xy + 2yz + 2xz\].
For two variables,
\[{\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab\]
\[{\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab\]