Question

Simplify the expression \[\left( 2x+5 \right)\left( 2x-5 \right)\]

Answer 1

Hint: In this type of question we have to use the concept of multiplication of algebraic expression. Here we can observe that the given question is in the form of \[\left( a+b \right)\left( a-b \right)\]. We know the algebraic identity which says that \[\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}\]. Using this identity in the given expression we can simplify it to the simplest form of the given expression.

Complete step-by-step solution:
Now we have to simplify the given expression \[\left( 2x+5 \right)\left( 2x-5 \right)\]
As the given expression is in the form of \[\left( a+b \right)\left( a-b \right)\] where \[a=2x\] and \[b=5\] we can use the algebraic identity, \[\left( a+b \right)\left( a-b \right)={{a}^{2}}-{{b}^{2}}\]
Hence, we can simplify the given expression as
\[\Rightarrow \left( 2x+5 \right)\left( 2x-5 \right)={{\left( 2x \right)}^{2}}-{{5}^{2}}\]
\[\Rightarrow \left( 2x+5 \right)\left( 2x-5 \right)={{\left( 2 \right)}^{2}}{{\left( x \right)}^{2}}-{{5}^{2}}\]
\[\Rightarrow \left( 2x+5 \right)\left( 2x-5 \right)=4{{x}^{2}}-25\]
Hence, by simplifying the above expression we get,
\[\left( 2x+5 \right)\left( 2x-5 \right)=4{{x}^{2}}-25\]

Note: In this type of question students have to use appropriate algebraic identity. One of the students may simplify the given expression by using distributive property also as follows:
By distributive property of multiplication of algebraic expressions we can write, \[\left( a+b \right)\left( c+d \right)=a\left( c+d \right)+b\left( c+d \right)\]
Here, we multiply the second expression by the first term of first expression and then the second expression by the second term of first expression.
Hence, we can simplify the above expression as
\[\Rightarrow \left( 2x+5 \right)\left( 2x-5 \right)=2x\left( 2x-5 \right)+5\left( 2x-5 \right)\]
By performing multiplication and simplifying each term we can write,
\[\Rightarrow \left( 2x+5 \right)\left( 2x-5 \right)={{2}^{2}}{{x}^{2}}-10x+10x-25\]
Now, by combining the like terms we get,
\[\Rightarrow \left( 2x+5 \right)\left( 2x-5 \right)=4{{x}^{2}}-25\]
Hence after simplification we get, \[\left( 2x+5 \right)\left( 2x-5 \right)=4{{x}^{2}}-25\]
During multiplication of algebraic expressions we have to note some of points like,
1. Multiplication of \['+'\] and \['-'\] gives us the \['-'\] sign always.
2. Always add same degree variables with variables and constant terms with constant terms.
3. BODMAS rule is the same as that of regular multiplication.