Question

Simplify:
\[\left( {3x + 2y - 9} \right)\left( {2x - 6y + 2} \right) - \left[ {\left( {4x - 9y - 1} \right) + \left( { - 3x + 8y + 7} \right)} \right]\]
A. \[6{x^2} - 14xy - 12{y^2} - 13x + 59y - 24\]
B. \[6{x^2} - 12xy - 189 - 17x + 61y - 29\]
C. \[8{x^2} - 14xy - 12{y^2} - 13x + 57y - 24\]
D. \[8{x^2} - 14xy - 12{y^2} - 17x + 61y - 29\]

Answer 1

Hint: Here we will solve this particular expression by using the property in which first of all we will solve the terms in the small brackets i.e. \[\left( {} \right)\] , after that we will solve all the terms inside the square brackets i.e. \[\left[ {} \right]\]. Finally we combine the similar terms and get the required answer.

Complete step-by-step solution:
Step 1: For solving the expression \[\left( {3x + 2y - 9} \right)\left( {2x - 6y + 2} \right) - \left[ {\left( {4x - 9y - 1} \right) + \left( { - 3x + 8y + 7} \right)} \right]\] , first of all, we will solve the small brackets from the right-hand side as shown below:
\[ \Rightarrow \left( {3x + 2y - 9} \right)\left( {2x - 6y + 2} \right) - \left[ {4x - 9y - 1 - 3x + 8y + 7} \right]\]
Step 2: Now simplifying the terms by doing simple addition and subtraction inside the \[\left[ {} \right]\]brackets on the RHS side of the above expression we get:
\[ \Rightarrow \left( {3x + 2y - 9} \right)\left( {2x - 6y + 2} \right) - \left[ {x - y + 6} \right]\]
Step 3: Now by multiplying the terms
\[\left( {3x + 2y - 9} \right)\left( {2x - 6y + 2} \right)\]we get:
\[ \Rightarrow \left( {6{x^2} - 18xy + 6x + 4xy - 12{y^2} + 4y - 18x + 54y - 18} \right) - \left[ {x - y + 6} \right]\]
Now after doing the simple addition and subtraction of the similar terms inside the small brackets of the above expression we get:
\[ \Rightarrow \left( {6{x^2} - 14xy - 12{y^2} - 12x + 58y - 18} \right) - \left[ {x - y + 6} \right]\]
Step 4: Now finally by opening both the brackets the operator sign inside the square brackets will change because there is a negative sign outside the brackets as shown below:
\[ \Rightarrow 6{x^2} - 14xy - 12{y^2} - 12x + 58y - 18 - x + y - 6\]
By doing the final addition and subtraction of similar terms, we get:
\[ \Rightarrow 6{x^2} - 14xy - 12{y^2} - 13x + 59y - 24\]

A is the correct option.

Note: Students need to remember the BODMAS rule for solving these types of questions. The full form of BODMAS is as given below:
B- Brackets
O- Orders
D- Division
M- Multiplication
A-Addition
S- Subtraction
Steps for solving the expression according to the BODMAS rule will be:
Calculations inside the brackets need to be solved first.
Orders or indices, which means any powers or root will be the next step for solving.
Before doing addition and subtraction, we will complete the division and multiplication part solving from left to right.
After that finally, we will solve the addition and subtraction from left to right to get the correct answer.