Question

Verify the validity of the identity: ${{\left( a+b+c \right)}^{2}}={{a}^{2}}+{{b}^{2}}+{{c}^{2}}+2ab+2bc+2ac$ for the value of $a=2,b=4,c=6$.

Answer 1

Hint: We can verify the given identity ${{\left( a+b+c \right)}^{2}}={{a}^{2}}+{{b}^{2}}+{{c}^{2}}+2ab+2bc+2ac$ by placing the given values of $a=2,b=4,c=6$. Then we get two expression values on both sides of the equality. If they match then the identity is verified.

Complete step-by-step answer:
To verify the validity of the identity ${{\left( a+b+c \right)}^{2}}={{a}^{2}}+{{b}^{2}}+{{c}^{2}}+2ab+2bc+2ac$ for the value of $a=2,b=4,c=6$, we need to place the values on both sides of the given identity. If the values match for both parts, then we can validate the given identity.
So, placing $a=2,b=4,c=6$ on the equation of ${{\left( a+b+c \right)}^{2}}$.
We have ${{\left( a+b+c \right)}^{2}}={{\left( 2+4+6 \right)}^{2}}={{12}^{2}}=144$.
Now we place $a=2,b=4,c=6$ on the expression of ${{a}^{2}}+{{b}^{2}}+{{c}^{2}}+2ab+2bc+2ac$.
$\begin{align}
  & {{a}^{2}}+{{b}^{2}}+{{c}^{2}}+2ab+2bc+2ac \\
 & ={{2}^{2}}+{{4}^{2}}+{{6}^{2}}+2\times 2\times 4+2\times 4\times 6+2\times 2\times 6 \\
 & =4+16+36+16+48+24 \\
 & =144 \\
\end{align}$
Thus, we verified the identity ${{\left( a+b+c \right)}^{2}}={{a}^{2}}+{{b}^{2}}+{{c}^{2}}+2ab+2bc+2ac$.

Note: The identity is derived from the simple multiplication of the two equal terms of ${{\left( a+b+c \right)}^{2}}$. We also can form it as the square of the sum of two numbers and find the identity.