Question

What is the product of $\left( 2x+5 \right)\left( 2x-5 \right)$?

Answer 1

Hint:Here we have to find the product of given algebraic expressions. Firstly we will write the algebraic identity $\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}$ then we will compare it with the algebraic expression given to us and get the value of two unknown variable. Then we will substitute the value obtained in the formula and simplify it to get our desired answer.

Complete step by step solution:
We have to find the product of the following expression:
$\left( 2x+5 \right)\left( 2x-5 \right)$…..$\left( 1 \right)$
We will use the algebraic identity given below:
$\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}$…..$\left( 2 \right)$
On comparing equation (1) by above identity we get,
$a=2x,b=5$
Substituting the values in equation (2) we get,
$\Rightarrow \left( 2x+5 \right)\left( 2x-5 \right)={{\left( 2x \right)}^{2}}-{{5}^{2}}$
$\Rightarrow \left( 2x+5 \right)\left( 2x-5 \right)=4{{x}^{2}}-25$
On simplifying we get the answer as $4{{x}^{2}}-25$ .
Hence the product of $\left( 2x+5 \right)\left( 2x-5 \right)$ is $4{{x}^{2}}-25$ .

Note:
Algebraic expressions are made up of variables and constants without an equal sign. Algebraic identities are the algebraic equation which is valid for all values of the variable in it. Algebraic equations are the equation with variable, constant and also an equal sign. In this type of question using the algebraic identities remove the work of long calculations and reduce the change of error. We can also find the product by multiplying each element of one bracket with each element of another bracket as follows:
$\Rightarrow \left( 2x+5 \right)\left( 2x-5 \right)=2x\times 2x+2x\times -5+5\times 2x+5\times -5$
$\Rightarrow \left( 2x+5 \right)\left( 2x-5 \right)=4{{x}^{2}}-10x+10x-25$
On simplifying further we get,
$\Rightarrow \left( 2x+5 \right)\left( 2x-5 \right)=4{{x}^{2}}+25$
We got the same answer.
It is easy to identify the values in which $\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}$ formula can be directly used as both the values in the two brackets are same with the middle sign as different. There are many other identities which are used widely in mathematics especially when we have to solve the algebraic identities.