Question

Expand the following:
\[{\left( {3a - 5b - c} \right)^2}\]

Answer 1

Hint: There is nothing much to do. First just expand the square, after expanding we have two terms. Multiply those two terms with each other. After multiplication we get an equation so arrange that equation by combining the like terms. After combining, add those terms. On further solving you will get the desired expression.

Complete step-by-step solution:
The given expression is \[{\left( {3a - 5b - c} \right)^2}\]. So we will now expand the given expression to make our next steps easier. On expanding the square the given expression becomes
\[ \Rightarrow \left( {3a - 5b - c} \right)\left( {3a - 5b - c} \right)\]
To move further we will now multiply each term of the first bracket with the second bracket
\[ \Rightarrow 3a\left( {3a - 5b - c} \right) - 5b\left( {3a - 5b - c} \right) - c\left( {3a - 5b - c} \right)\]
So expand it further by doing multiplication. Therefore the above expression becomes
\[ \Rightarrow 9{a^2} - 15ab - 3ac - 15ab + 25{b^2} + 5bc - 3ac + 5bc + {c^2}\]
Now write the above equation in its correct way by combining the like terms present in the equation to make our calculations or we can say to make our next steps easier
\[ \Rightarrow 9{a^2} + 25{b^2} + {c^2} - 15ab - 15ab + 5bc + 5bc - 3ac - 3ac\]
So finally the equation is in its correct way. Now add the like terms so that the like terms will reduce. By doing this we get
\[ \Rightarrow 9{a^2} + 25{b^2} + {c^2} - 30ab + 10bc - 6ac\]
Hence this is our final answer to the given question. That is on expanding \[{\left( {3a - 5b - c} \right)^2}\] we get \[9{a^2} + 25{b^2} + {c^2} - 30ab + 10bc - 6ac\]

Note: Keep in mind that if you want to recheck your answer than you can use the identity \[{\left( {a + b + c} \right)^2} = {a^2} + {b^2} + {c^2} + 2ab + 2bc + 2ca\]. By putting the values according to the given expression in this identity you can check if your answer is right or wrong. This is the easiest way to check or we can say that this is the alternative method to find the answer.