Question

Solve the following:
$\left( 7x+3 \right)\left( 7x-3 \right)$

Answer 1

Hint: Here we have been given an algebraic expression and we have to solve it. So we will use algebraic identity to solve the given expression. Firstly we will write the algebraic identity ${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)$ and compare it with the expression given and get the value of the $a,b$ . Then substitute the value in this identity and get the desired answer.

Complete step by step answer:
The algebraic expression is given as follows:
$\left( 7x+3 \right)\left( 7x-3 \right)$….$\left( 1 \right)$
Now we know the algebraic identity as follows:
${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)$
On comparing equation (1) by the right side of the above identity we get,
$a=7x$ And $b=3$
On substituting the above value in the left side of the identity and simplify we get,
$\Rightarrow \left( {{\left( 7x \right)}^{2}}-{{\left( 3 \right)}^{2}} \right)=\left( 7x+3 \right)\left( 7x-3 \right)$
$\Rightarrow \left( 49{{x}^{2}}-9 \right)=\left( 7x+3 \right)\left( 7x-3 \right)$
So we get the value from the above expression as $49{{x}^{2}}-9$
Hence on solving $\left( 7x+3 \right)\left( 7x-3 \right)$ we get the solution as $49{{x}^{2}}-9$ .

Note:
Algebraic identities are usually used to solve this type of questions but if we are not able to figure out which identity can be used we can simplify and multiply the two brackets. Multiply each term of the first bracket to each term of the second brackets as follows,
$\left( 7x+3 \right)\left( 7x-3 \right)$
$\Rightarrow \left( 7x+3 \right)\left( 7x-3 \right)=\left( 7x\times 7x \right)+\left( 7x\times -3 \right)+\left( 3\times 7x \right)+\left( 3\times -3 \right)$
$\Rightarrow \left( 7x+3 \right)\left( 7x-3 \right)=14{{x}^{2}}-21x+21x-9$
So we get the answer as,
$\Rightarrow \left( 7x+3 \right)\left( 7x-3 \right)=14{{x}^{2}}-9$
So we get the same answer.
The identities are used to reduce our calculation process where chances of error are more. Here the numbers are small so we can simply multiply them but when the values are bigger using this approach is complicated. Algebraic identities are those algebraic equations that are valid for all values of variables in it. They are generally used in factoring the polynomials and solving the expressions.