Question

How do you expand ${(x - 2)^2}$?

Answer 1

Hint: The algebraic expression given in the question can be expanded with the help of algebraic identities. Let us first know what an algebraic identity is. It is an algebraic equation which holds true for all the values of its variable. There are four important algebraic identities as given below:
5) ${(a + b)^2} = {a^2} + 2ab + {b^2}$
6) ${(a - b)^2} = {a^2} - 2ab + {b^2}$
7) ${a^2} - {b^2} = (a + b)(a - b)$
8) $(x + a)(x + b) = {x^2} + x(a + b) + ab$
Here to solve the given question, we will use the algebraic identity ${(a - b)^2} = {a^2} - 2ab + {b^2}$.

Complete step by step solution:
The given algebraic expression is ${(x - 2)^2}$. It is in the form of ${(a - b)^2}$ in the identity ${(a - b)^2} = {a^2} - 2ab + {b^2}$. So now we will find out $a$ and $b$, such that
$ \Rightarrow a = x$ and $ \Rightarrow b = 2$.
Hence, we have $a = x$ and $b = 2$. On substituting these values on the right hand side of the identity ${(a - b)^2} = {a^2} - 2ab + {b^2}$, we will get
$ \Rightarrow {(a - b)^2} = {a^2} - 2ab + {b^2}$
$ \Rightarrow {(x - 2)^2} = {(x)^2} - (2 \times x \times 2) + {(2)^2}$
On simplifying the right hand side of the equation, we will get
$ \Rightarrow {(x - 2)^2} = {x^2} - 4x + 4$

Hence, when we expand ${(x - 2)^2}$, we get ${x^2} - 4x + 4$ as the answer.

Note:
The given question could have also been solved by transforming ${(x - 2)^2}$ as $[{(x + ( - 2)]^2}$ and then using the identity ${(a + b)^2} = {a^2} + 2ab + {b^2}$.
In this case, we will have $a = x$ and $b = - 2$. On substituting these values on the right hand side of the identity ${(a + b)^2} = {a^2} + 2ab + {b^2}$, we will get
$ \Rightarrow {(a + b)^2} = {a^2} + 2ab + {b^2}$
$ \Rightarrow {[x + ( - 2)]^2} = [{(x)^2} + (2 \times x \times - 2) + {( - 2)^2}]$
On simplifying the right hand side of the equation, we will get
$ \Rightarrow {[x + ( - 2)]^2} = [{x^2} - (2 \times x \times 2) + 4]$
$ \Rightarrow {[x + ( - 2)]^2} = {x^2} - 4x + 4$
Hence, when we expand $[{(x + ( - 2)]^2}$, we get ${x^2} - 4x + 4$ as the answer.
Moreover, two of the mentioned identities given are ${(a + b)^2} = {a^2} + 2ab + {b^2}$ and ${(a - b)^2} = {a^2} - 2ab + {b^2}$. These two identities are further used in another identity which is also important, i.e.
${(a + b)^2} - {(a - b)^2} = 4ab$.