Question

Find the square of ${\left( {2a + b} \right)^2}$.

Answer 1

Hint: In the given question, we are required to find the value of the square of the expression ${\left( {2a + b} \right)^2}$. Given is a bracket with an expression involving two variables, a and b. So, we have to evaluate the square of the given term. Square is nothing but multiplying the same number with itself. We can find the square of the expression using the algebraic identity ${\left( {x + y} \right)^2} = {x^2} + 2xy + {y^2}$.

Complete step-by-step solution:
In the given question, we have to find the square of the algebraic expression $\left( {2a + b} \right)$. So, we will find the square using the algebraic identity ${\left( {x + y} \right)^2} = {x^2} + 2xy + {y^2}$. The identity is used for finding the sum of a binomial expression.
So, we have, ${\left( {2a + b} \right)^2}$
Using the algebraic identity ${\left( {x + y} \right)^2} = {x^2} + 2xy + {y^2}$, we get,
${\left( {2a + b} \right)^2} = {\left( {2a} \right)^2} + 2\left( {2a} \right)\left( b \right) + {\left( b \right)^2}$
Now, simplifying the expression by computing the square of the terms, we get,
${\left( {2a + b} \right)^2} = 4{a^2} + 2\left( {2a} \right)\left( b \right) + {b^2}$
Now, computing the required multiplication, we get,
${\left( {2a + b} \right)^2} = 4{a^2} + 4ab + {b^2}$
So, the square of the expression ${\left( {2a + b} \right)^2}$ is equal to $4{a^2} + 4ab + {b^2}$.
Hence, option (D) is the correct answer.

Note: We can also find the square of the given expression by multiplying the expression with itself.
So, we get, ${\left( {2a + b} \right)^2} = \left( {2a + b} \right)\left( {2a + b} \right)$
Now, we will multiply the first term of the first bracket with the other bracket and then the second term of the first bracket with the whole bracket. So, we get,
${\left( {2a + b} \right)^2} = 2a\left( {2a + b} \right) + b\left( {2a + b} \right)$
Multiplying the terms and simplifying the expression, we get,
${\left( {2a + b} \right)^2} = 4{a^2} + 2ab + 2ab + {b^2}$
Now, adding up the similar terms, we get,
${\left( {2a + b} \right)^2} = 4{a^2} + 4ab + {b^2}$
So, we get the same final answer using both the methods. We must remember the algebraic identities to solve such questions.