The equation ${\left( {a + b} \right)^2} = 4ab$ is possible when:
(A) $2a = b$
(B) $3a = 2b$
(C) $a = b$
(D) $a = 2b$

Hint: Apply the algebraic formula ${\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab$ in the left hand side of the equation. Then simplify it further to establish a relation between $a$ and $b$ and eventually the answer.

Complete step-by-step solution:
According to the question, the given equation is ${\left( {a + b} \right)^2} = 4ab$.
We know that the algebraic formula for ${\left( {a + b} \right)^2}$ is given as:
\[ \Rightarrow {\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab\]
Using this formula, we’ll get:
$ \Rightarrow {a^2} + {b^2} + 2ab = 4ab$
On further simplification, we’ll get:
\[
   \Rightarrow {a^2} + {b^2} + 2ab - 4ab = 0 \\
   \Rightarrow {a^2} + {b^2} - 2ab = 0 \\
 \]
Further, according to another algebraic formula, we have:
\[ \Rightarrow {\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab\]
Again using this formula for the above resolved equation, we’ll get:
$
   \Rightarrow {\left( {a - b} \right)^2} = 0 \\
   \Rightarrow a - b = 0 \\
   \Rightarrow a = b \\
 $
Thus for the above equation to be satisfied, the condition that must be followed is $a = b$.
(C) is the correct option.

Additional Information: The square-root of ${\left( {a - b} \right)^2}$ taken in the above step of the solution should come with modulus sign as:
$ \Rightarrow \sqrt {{{\left( {a - b} \right)}^2}} = \left| {a - b} \right|$
But since the right hand side in the equation is zero so it doesn’t matter whether we are taking the modulus sign or not.
\[ \Rightarrow {\left( {a - b} \right)^2} = 0\]
Taking square-root both sides, we’ll get:
\[
   \Rightarrow \left| {a - b} \right| = 0 \\
   \Rightarrow a - b = 0
 \]

Note: Some of the other frequently used algebraic formulas are:
$
   \Rightarrow \left( {a + b} \right)\left( {a - b} \right) = {a^2} - {b^2} \\
   \Rightarrow {\left( {a + b} \right)^3} = {a^3} + {b^3} + 3ab\left( {a + b} \right) \\
   \Rightarrow {\left( {a - b} \right)^3} = {a^3} - {b^3} - 3ab\left( {a - b} \right) \\
   \Rightarrow {a^3} + {b^3} = \left( {a + b} \right)\left( {{a^2} + {b^2} - ab} \right) \\
   \Rightarrow {a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + {b^2} + ab} \right)
 $