Question

Expansion of \[{\left( {2a + 3b + 4c} \right)^2}\] is ______
A.\[4{a^2} + 9{b^2} + 16{c^2} + 12ab + 24bc + 16ca\]
B. \[{a^2} + {b^2} + 2ca\]
C. \[{a^2} - {b^2} - 2ca\]
D. \[{a^2} - {b^2} + 2ca\]

Answer 1

Hint: We use the formula of expansion of square of sum of three numbers and substitute the required values as per the question. Cancel the terms with the same magnitude but opposite sign and reduce the answer to the shortest and simplest possible form.
* If x, y and z are three numbers then \[{\left( {x + y + z} \right)^2} = {x^2} + {y^2} + {z^2} + 2xy + 2yz + 2zx\]

Complete step-by-step solution:
We have to find the value of \[{\left( {2a + 3b + 4c} \right)^2}\]
Since the expansion \[{\left( {2a + 3b + 4c} \right)^2}\] means the square of sum of three numbers ‘2a’, ‘3b’ and ‘4c’, we use the formula of expansion of square of sum of three numbers and compare the expansion with general expansion.
Compare the three numbers with numbers in the general expansion formula \[{\left( {x + y + z} \right)^2}\]
On comparing with general form of numbers we get \[x = 2a,y = 3b,z = 4c\]
Substitute the values of x, y and z in the general formula and calculate the value
\[ \Rightarrow {\left( {2a + 3b + 4c} \right)^2} = {\left( {2a} \right)^2} + {\left( {3b} \right)^2} + {\left( {4c} \right)^2} + 2\left( {2a \times 3b} \right) + 2\left( {3b \times 4c} \right) + 2\left( {4c \times 2a} \right)\]
Calculate the square terms in RHS if the equation and write the value of products inside brackets in RHS
\[ \Rightarrow {\left( {2a + 3b + 4c} \right)^2} = 2a \times 2a + 3b \times 3b + 4c \times 4c + 2\left( {6ab} \right) + 2\left( {12bc} \right) + 2\left( {8ac} \right)\]
We know when base is same we can add the power of the numbers
\[ \Rightarrow {\left( {2a + 3b + 4c} \right)^2} = 4{a^2} + 9{b^2} + 16{c^2} + 2\left( {6ab} \right) + 2\left( {12bc} \right) + 2\left( {8ac} \right)\]
Multiply the terms in bracket to the term outside the bracket
\[ \Rightarrow {\left( {2a + 3b + 4c} \right)^2} = 4{a^2} + 9{b^2} + 16{c^2} + 12ab + 24bc + 16ac\]
\[\therefore \]The value of \[{\left( {2a + 3b + 4c} \right)^2}\] is \[4{a^2} + 9{b^2} + 16{c^2} + 12ab + 24bc + 16ac\]

\[\therefore \]Option A is correct.

Note: Students many times make the mistake of collecting the common factor between the terms and start grouping the terms together on the basis of that common term. Keep in mind here we have to match the answer with the given options so we calculate the expansion very carefully according to the options. Students are advised to calculate the square of the given numbers in the expansion by writing the product of the same number with it and then adding the powers of terms having the same base to avoid confusion.