Question

Solve:
${{\left( 3a-b \right)}^{2}}-4\left( 3a-b \right)+4$

Answer 1

Hint: Here we have been given an expression and we have to solve it. Firstly as we can see that by opening the bracket we will not be able to solve it. So we will use algebraic identities for solving it. We will write the formula for the square of difference of two values and then by comparing it by the expression given, rewrite the expression and get our desired answer.

Complete step by step answer:
The expression is given as follows,
${{\left( 3a-b \right)}^{2}}-4\left( 3a-b \right)+4$
We rewrite the above expression by breaking the value $4$ as $2\times 2$ and ${{2}^{2}}$ as follows,
${{\left( 3a-b \right)}^{2}}-2\times 2\times \left( 3a-b \right)+{{2}^{2}}$ …..$\left( 1 \right)$
Now as we know the algebraic identity as follows:
${{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$
On comparing equation (1) right side by above identity we get,
$a=3a-b$ And $b=2$
Now using identity we get,
$\Rightarrow {{\left( 3a-b-2 \right)}^{2}}={{\left( 3a-b \right)}^{2}}-2\times 2\left( 3a-b \right)+{{2}^{2}}$
So we get the expression as,
$\Rightarrow {{\left( 3a-b-2 \right)}^{2}}={{\left( 3a-b \right)}^{2}}-4\left( 3a-b \right)+4$
Hence on solving ${{\left( 3a-b \right)}^{2}}-4\left( 3a-b \right)+4$ we get the solution as ${{\left( 3a-b-2 \right)}^{2}}$ .

Note:
Algebraic identities are those algebraic equations which are valid for all values of the variables in it. They are used to solve different polynomials by factoring them. As it is not given how to solve the expression given we can simply open the bracket in the expression and simplify it to get the solution as follows,
${{\left( 3a-b \right)}^{2}}-4\left( 3a-b \right)+4$
Using below formula in first term of above expression,
${{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$
We get,
$\Rightarrow \left( {{\left( 3a \right)}^{2}}-2\times 3a\times b+{{b}^{2}} \right)-4\left( 3a-b \right)+4$
$\Rightarrow \left( 9{{a}^{2}}-6ab+{{b}^{2}} \right)-12a+4b+4$
Simplifying further we get,
$\Rightarrow 9{{a}^{2}}-6ab+{{b}^{2}}-12a+4b+4$
So our solution is $9{{a}^{2}}-6ab+{{b}^{2}}-12a+4b+4$ .
We can solve our expression as above but it is not giving us a compact form of answer so using the algebraic identity and finding a compact answer is more preferable in such questions.