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How do you multiply \[{(x - 2y)^2}\]?

Last updated date: 11th Jun 2024
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Hint: Here in this question we have to simplify the given algebraic equation. To simplify the above algebraic equation we use arithmetic operation multiplication. It is also simplified by using the standard algebraic formula \[{(a - b)^2} = {a^2} - 2ab + {b^2}\]. Hence we can obtain the solution for the question.

Complete step-by-step solution:
The algebraic expression or equation is a combination of variables, constants and arithmetic operations.
The equation can be simplified by using two methods.

Method 1:
Now consider the given equation \[{(x - 2y)^2}\]
The exponential form can be expanded
\[ \Rightarrow (x - 2y)(x - 2y)\]
The terms in the braces are multiplied. On multiplying we get
\[ \Rightarrow x(x - 2y) - 2y(x - 2y)\]
On term-by-term multiplication
\[ \Rightarrow {x^2} - 2xy - 2xy + 4{y^2}\]
On simplifying we get
\[{x^2} - 4xy + 4{y^2}\]
Hence we have simplified the given equation

Method 2:
We solve or simplify the given algebraic equation by using the standard algebraic formula \[{(a - b)^2} = {a^2} - 2ab + {b^2}\].
Now consider the given equation \[{(x - 2y)^2}\]. On comparing the standard algebraic formula and the given equation. The value of a is x and the value of b is 2y.
On substituting these values in the algebraic formula we get
\[ \Rightarrow {(x - 2y)^2} = {(x)^2} - 2(x)(2y) + {(2y)^2}\]
On squaring and simplifying the terms we get
\[ \Rightarrow {(x - 2y)^2} = {x^2} - 4xy + 4{y^2}\]
Hence we have simplified the given equation
Solving the equation by method 1 and method 2 we have obtained the final answer as the same.
Therefore \[{(x - 2y)^2} = {x^2} - 4xy + 4{y^2}\]

Note: To multiply we use operation multiplication, multiplication of numbers is different from the multiplication of algebraic expression. In the algebraic expression it involves the both number that is constant and variables. Variables are also multiplied, if the variable is the same then the result will be in the form of exponent. We must know about the standard algebraic formulas.