Question

How do you find the product of \[{{\left( x+6 \right)}^{2}}\]?

Answer 1

Hint: To solve the given expression we should know the expansion of expression, \[\left( a+b \right)\left( c+d \right)\]. The expansion is done by multiplying the second bracket with the terms in the first bracket. And adding their product. After this, the second bracket is expanded, and we get the result as the product of all the terms in the first bracket with the terms in the second bracket. This can be expressed algebraically as,
\[\begin{align}
  & \left( a+b \right)\left( c+d \right) \\
 & \Rightarrow a\left( c+d \right)+b\left( c+d \right) \\
 & \Rightarrow ac+ad+bc+bd \\
\end{align}\]

Complete answer:
We are given the expression \[{{\left( x+6 \right)}^{2}}\], to find the product we first have to express it as the product form. As the given expression has the power 2. So, it can be written as the product of \[\left( x+6 \right)\] with itself. By doing this we get,
\[\Rightarrow {{\left( x+6 \right)}^{2}}=\left( x+6 \right)\left( x+6 \right)\]
The above expression is of the form \[\left( a+b \right)\left( c+d \right)\], we know that the expansion of this expression is written as, \[a\left( c+d \right)+b\left( c+d \right)\]. So, we can expand the expression \[\left( x+6 \right)\left( x+6 \right)\] similarly. By doing this we get,
\[\begin{align}
  & \Rightarrow \left( x+6 \right)\left( x+6 \right) \\
 & \Rightarrow x\left( x+6 \right)+6\left( x+6 \right) \\
\end{align}\]
Expanding the bracket terms, we get
\[\begin{align}
  & \Rightarrow x\left( x+6 \right)+6\left( x+6 \right) \\
 & \Rightarrow x\times x+6\times x+6\times x+6\times 6 \\
\end{align}\]
\[\Rightarrow {{x}^{2}}+6x+6x+36\]
The above expression can also be written as,
\[\Rightarrow {{x}^{2}}+12x+36\]
Hence the product of \[{{\left( x+6 \right)}^{2}}\], which can also be written as \[\left( x+6 \right)\left( x+6 \right)\] equals \[{{x}^{2}}+12x+36\].

Note: The given problem can also be done by remembering the expansion formula of the expression of the form \[{{\left( a+b \right)}^{2}}\]. The expansion of this expression is \[{{a}^{2}}+2ab+{{b}^{2}}\]. As the given expression is of the form \[{{\left( a+b \right)}^{2}}\], we can expand it by simply substituting the values of a, and b. on comparing we get that, a = x, and b = 6.