Question

Simplify ${\left( {x - 2y} \right)^2} - 4x + 8y$

Answer 1

Hint: In the given question, we have to simplify an algebraic expression given to us in the problem itself. The whole square term can be simplified with the help of algebraic identity ${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$. The algebraic identity ${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$ is used to evaluate the square of a binomial expression involving the difference of two terms.

Complete step-by-step solution:
Given question requires us to simplify the algebraic expression ${\left( {x - 2y} \right)^2} - 4x + 8y$.
So, in order to simplify the algebraic expression ${\left( {x - 2y} \right)^2} - 4x + 8y$, we first evaluate the whole square bracket ${\left( {x - 2y} \right)^2}$.
So, to calculate the square of the term $\left( {x - 2y} \right)$, we have to first recognise the two parts of the binomial and then substitute the values in the algebraic identity.
So, the two terms in the binomial $\left( {x - 2y} \right)$ are $x$ and $2y$.
Hence, we have, ${\left( {x - 2y} \right)^2} = {x^2} - 2\left( x \right)\left( {2y} \right) + {\left( {2y} \right)^2}$
Simplifying the expression further, we get,
$ \Rightarrow {\left( {x - 2y} \right)^2} = {x^2} - 4xy + 4{y^2}$
Now, we put the value of square of $\left( {x - 2y} \right)$ into the expression ${\left( {x - 2y} \right)^2} - 4x + 8y$. So, we get,
${\left( {x - 2y} \right)^2} - 4x + 8y = {x^2} - 4xy + 4{y^2} - 4x + 8y$

So, the simplified form of the expression ${\left( {x - 2y} \right)^2} - 4x + 8y$ is ${x^2} - 4xy + 4{y^2} - 4x + 8y$.

Note: Before attempting such questions, one should memorize all the algebraic identities and should know their applications in such problems. Care should be taken while carrying out the calculations. The answer for the given question can also be verified by actually multiplying the term $\left( {x - 2y} \right)$ twice in order to calculate the square.