Question

Prove that \[{{\left( a+b \right)}^{2}}-{{\left( a-b \right)}^{2}}=4ab\].

Answer 1

Hint: To prove the given equation, expand the terms given on the left side of the given equation using the identities \[{{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab\] and \[{{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab\]. Simplify the equation to prove the given equation.

Complete step-by-step answer:
We have to prove that \[{{\left( a+b \right)}^{2}}-{{\left( a-b \right)}^{2}}=4ab\].
To do so, we will expand the terms on the left side of the given equation.
We know that \[{{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab\] and \[{{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab\].
Thus, we have \[{{\left( a+b \right)}^{2}}-{{\left( a-b \right)}^{2}}=\left( {{a}^{2}}+{{b}^{2}}+2ab \right)-\left( {{a}^{2}}+{{b}^{2}}-2ab \right)\].
Simplifying the above equation, we have \[{{\left( a+b \right)}^{2}}-{{\left( a-b \right)}^{2}}=2ab-\left( -2ab \right)=4ab\].
Hence, we have proved the given equation \[{{\left( a+b \right)}^{2}}-{{\left( a-b \right)}^{2}}=4ab\].
All these identities used in this question are algebraic identities. An algebraic identity is an equality that holds for all possible values of its variables. We can prove each of the identities by performing basic algebraic operations such as addition, multiplication, subtraction and division. They are used for the factorization of the polynomials. That’s why they are useful in the computation of algebraic expressions. An algebraic expression differs from an algebraic identity in the way that the value of an algebraic expression changes with the change in variables. However, an algebraic identity is an equality which holds for all possible values of variables.
We can verify an algebraic identity using the substitution method. In this method, we substitute the values for the variables and perform arithmetic operations. We can also prove an algebraic identity using an activity method, in which we use geometry to prove the algebraic identity.

Note: We can also solve this question by using another algebraic identity \[{{x}^{2}}-{{y}^{2}}=\left( x+y \right)\left( x-y \right)\]. Substitute \[x=a+b,y=a-b\] in the above equation to write \[{{\left( a+b \right)}^{2}}-{{\left( a-b \right)}^{2}}=\left[ \left\{ \left( a+b \right)+\left( a-b \right) \right\}\left\{ \left( a+b \right)-\left( a-b \right) \right\} \right]\]. Further simplifying the previous equation, we have \[{{\left( a+b \right)}^{2}}-{{\left( a-b \right)}^{2}}=\left( 2a \right)\left( 2b \right)\]. Thus, we have \[{{\left( a+b \right)}^{2}}-{{\left( a-b \right)}^{2}}=4ab\].