Question

Simplify: \[{\left( {2x + 3} \right)^3} - {\left( {2x - 3} \right)^3}\]
A) \[54 + 72{x^2}\]
B) \[54 + 18{x^2}\]
C) \[54 + 72{x^3}\]
D) \[54x + 72{x^2}\]

Answer 1

Hint:
To solve the given expression, we will first apply the formula of cube of sum of two terms to find the expansion of \[{\left( {2x + 3} \right)^3}\]. Then we will use the formula of cube of difference of two terms to find the expansion of \[{\left( {2x - 3} \right)^3}\]. We will then find their differences to get the required answer.

Complete step by step solution:
Let \[S = {\left( {2x + 3} \right)^3} - {\left( {2x - 3} \right)^3}\] …… \[\left( 1 \right)\]
We will first find the expansion of \[{\left( {2x + 3} \right)^3}\] using the formula of cube of sum of two terms.
Using the algebraic identity, \[{\left( {a + b} \right)^3} = {a^3} + {b^3} + 3{a^2}b + 3a{b^2}\] , we can write
\[{\left( {2x + 3} \right)^3} = {\left( {2x} \right)^3} + {3^3} + 3 \times {\left( {2x} \right)^2} \times 3 + 3 \times 2x \times {3^2}\]
Applying exponents on the bases, we get
\[ \Rightarrow {\left( {2x + 3} \right)^3} = 8{x^3} + 27 + 3 \times 4{x^2} \times 3 + 3 \times 2x \times 9\]
Multiplying the terms, we get
\[ \Rightarrow {\left( {2x + 3} \right)^3} = 8{x^3} + 27 + 36{x^2} + 54x\] ……. \[\left( 2 \right)\]
Similarly, we will first find the expansion of \[{\left( {2x - 3} \right)^3}\] using the formula of the cube of difference of two terms.
Using the algebraic identity, \[{\left( {a - b} \right)^3} = {a^3} - {b^3} - 3{a^2}b + 3a{b^2}\] , we get
\[ \Rightarrow {\left( {2x - 3} \right)^3} = {\left( {2x} \right)^3} - {3^3} - 3 \times {\left( {2x} \right)^2} \times 3 + 3 \times 2x \times {3^2}\]
Applying exponents on the bases, we get
\[ \Rightarrow {\left( {2x - 3} \right)^3} = 8{x^3} - 27 - 3 \times 4{x^2} \times 3 + 3 \times 2x \times 9\]
Multiplying the terms, we get
\[ \Rightarrow {\left( {2x - 3} \right)^3} = 8{x^3} - 27 - 36{x^2} + 54x\] ……. \[\left( 3 \right)\]
Substituting the value of \[{\left( {2x + 3} \right)^3}\] and \[{\left( {2x - 3} \right)^3}\] obtained in equation \[\left( 2 \right)\] and \[\left( 3 \right)\] in equation \[\left( 1 \right)\], we get
\[ \Rightarrow S = 8{x^3} + 27 + 36{x^2} + 54x - \left( {8{x^3} - 27 - 36{x^2} + 54x} \right)\]
Opening the bracket by multiplying \[ - 1\] to each term inside the brackets, we get
\[ \Rightarrow S = 8{x^3} + 27 + 36{x^2} + 54x - 8{x^3} + 27 + 36{x^2} - 54x\]
Subtracting and adding the like terms, we get
\[ \Rightarrow S = 54 + 72{x^2}\]

Hence, the correct option is option A.

Note:
We can also solve the question using an alternate method. We will use the formula \[{a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + ab + {b^2}} \right)\].
Now we can write the given equation, as
\[{\left( {2x + 3} \right)^3} - {\left( {2x - 3} \right)^3} = \left( {\left( {2x + 3} \right) - \left( {2x - 3} \right)} \right)\left( {{{\left( {2x + 3} \right)}^2} + \left( {2x + 3} \right)\left( {2x - 3} \right) + {{\left( {2x - 3} \right)}^2}} \right)\]
Simplifying the terms, we get
\[ \Rightarrow {\left( {2x + 3} \right)^3} - {\left( {2x - 3} \right)^3} = \left( {2x + 3 - 2x + 3} \right)\left( {4{x^2} + 9 + 12x + 4x - 9 + 4{x^2} + 9 - 12x} \right)\]
Cancelling out like terms, we get
\[ \Rightarrow {\left( {2x + 3} \right)^3} - {\left( {2x - 3} \right)^3} = \left( {3 + 3} \right)\left( {4{x^2} + 4x + 4{x^2} + 9} \right)\]
Adding the like terms, we get
\[ \Rightarrow {\left( {2x + 3} \right)^3} - {\left( {2x - 3} \right)^3} = \left( 6 \right)\left( {12{x^2} + 9} \right)\]
Multiplying the terms using distributive property, we get
\[ \Rightarrow {\left( {2x + 3} \right)^3} - {\left( {2x - 3} \right)^3} = 72{x^2} + 54\]