Question

Write the expanded form of ${{\left( a+2b+c \right)}^{2}}$ .

Answer 1

Hint: To write the expanded form of the given expression, we have to use the algebraic identity ${{\left( a+b+c \right)}^{2}}={{a}^{2}}+{{b}^{2}}+{{c}^{2}}+2ab+2bc+2ac$ . We have to compare the terms of the given expression and this identity, substitute the values and simplify.

Complete step by step answer:
We have to write the expanded form of ${{\left( a+2b+c \right)}^{2}}$ . We know that ${{\left( a+b+c \right)}^{2}}={{a}^{2}}+{{b}^{2}}+{{c}^{2}}+2ab+2bc+2ac$
Let us compare the given expression with the above formula. We have to substitute for b as 2b in the identity for ${{\left( a+b+c \right)}^{2}}$ .
$\Rightarrow {{\left( a+2b+c \right)}^{2}}={{a}^{2}}+{{\left( 2b \right)}^{2}}+{{c}^{2}}+2a\times 2b+2\times 2b\times c+2ac$
Let us simplify the above expression.
$\Rightarrow {{\left( a+2b+c \right)}^{2}}={{a}^{2}}+4{{b}^{2}}+{{c}^{2}}+4ab+4bc+2ac$
Hence, the expanded form of ${{\left( a+2b+c \right)}^{2}}$ is ${{a}^{2}}+4{{b}^{2}}+{{c}^{2}}+4ab+4bc+2ac$ .

Note: Students must be thorough with algebraic identities. They have a chance of making mistake by writing the identity as ${{\left( a+b+c \right)}^{2}}={{a}^{2}}+{{b}^{2}}+{{c}^{2}}+4ab+4bc+4ac$ . The main identities used in algebra are that of ${{\left( a+b \right)}^{2}},{{\left( a-b \right)}^{2}},\left( {{a}^{2}}-{{b}^{2}} \right),{{\left( a+b+c \right)}^{2}},{{\left( a-b-c \right)}^{2}}$ . They can also use the identity ${{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}$ to find the value of the given expression.
Let us group the given expression as ${{\left( \left( a+2b \right)+c \right)}^{2}}$ . Now, we have to substitute $a=a+2b$ and $b=c$ in the identity for ${{\left( a+b \right)}^{2}}$ .
$\Rightarrow {{\left( \left( a+2b \right)+c \right)}^{2}}={{\left( a+2b \right)}^{2}}+2\left( a+2b \right)c+{{c}^{2}}$
Again, we have to apply the identity ${{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}$ on the first term.
$\Rightarrow {{\left( \left( a+2b \right)+c \right)}^{2}}={{a}^{2}}+2a\times 2b+{{\left( 2b \right)}^{2}}+2\left( a+2b \right)c+{{c}^{2}}$
Let us simplify the above expression using distributive property.
\[\begin{align}
  & \Rightarrow {{\left( \left( a+2b \right)+c \right)}^{2}}={{a}^{2}}+4ab+4{{b}^{2}}+2ac+4bc+{{c}^{2}} \\
 & \Rightarrow {{\left( \left( a+2b \right)+c \right)}^{2}}={{a}^{2}}+4{{b}^{2}}+{{c}^{2}}+4ab+4bc+2ac \\
\end{align}\]
We can also combine the terms ${{\left( a+\left( 2b+c \right) \right)}^{2}}$ and substitute $b=2b+c$ in the identity ${{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}$ .
$\Rightarrow {{\left( a+\left( 2b+c \right) \right)}^{2}}={{a}^{2}}+2a\times \left( 2b+c \right)+{{\left( 2b+c \right)}^{2}}$
Now, let us apply the identity ${{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}$ again on the third term of the above expression.
$\Rightarrow {{\left( a+\left( 2b+c \right) \right)}^{2}}={{a}^{2}}+2a\times \left( 2b+c \right)+{{\left( 2b \right)}^{2}}+2\times 2b\times c+{{c}^{2}}$
Let us simplify the above expression using distributive property.
\[\begin{align}
  & \Rightarrow {{\left( a+\left( 2b+c \right) \right)}^{2}}={{a}^{2}}+4ab+2ac+4{{b}^{2}}+4bc+{{c}^{2}} \\
 & \Rightarrow {{\left( a+\left( 2b+c \right) \right)}^{2}}={{a}^{2}}+4{{b}^{2}}+{{c}^{2}}+4ab+4bc+2ac \\
\end{align}\]
We can see that these methods are somewhat lengthy. Therefore, students must learn the formulas to save time.