Question

How do you find the product ${\left( {x - 6} \right)^2}$?

Answer 1

We will use the method of multiplication of variables to solve this question. So, when variables are the same, multiplying them together compresses them into a single factor (variable). When multiplying variables, we multiply the coefficients and variables as usual. If the bases are the same, we can multiply the bases by merely adding their exponents.

Complete step-by-step answer:
Before proceeding with the question, we should understand the concept of multiplication of variables.
If we are using one variable and one constant, then all we have to do is write the constant and variable together without the multiplication sign. Example: $x \times 2 = 2x$, this occurs because x and 2 represent two different amounts.
If we are multiplying a variable with itself then it is simply that variable squared or cubed or whatever power depending upon how many times you multiplied that variable by itself. Example: $x \times x = {x^2}$.
$\Rightarrow {\left( {x - 6} \right)^2} = \left( {x - 6} \right)\left( {x - 6} \right)$
As we know that $x \times x = {x^2}$ and any constant multiplied by the variable x is simply constant times the variable x. So, applying these we get,
$\Rightarrow {\left( {x - 6} \right)^2} = x \times x - 6 \times x - 6 \times x + 6 \times 6$
Now simplifying and rearranging equation (2) we get,
$\Rightarrow {\left( {x - 6} \right)^2} = {x^2} - 12x + 36$

Hence, the product of ${\left( {x - 6} \right)^2}$ is ${x^2} - 12x + 36$.

Note:
This can be done in another way also.
We know that the algebraic identity,
${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$
Using the above identity where $a = x$ and $b = 6$, we get
$\Rightarrow {\left( {x - 6} \right)^2} = {x^2} - 2 \times x \times 6 + {6^2}$
Simplify the terms,
$\Rightarrow {\left( {x - 6} \right)^2} = {x^2} - 12x + 36$
We have to remember the basic definition of multiplication of different variables and of multiplication of the same variables to solve this type of question. We in a hurry may commit a mistake in the multiplication of terms, so we need to be careful with this.