%5E2%3Da%5E2%2B2ab%2Bb%5E2-proof.jpeg "(a+b)^2=a^2+2ab+b^2 Proof")
(a+b)^2 = a^2 + 2ab + b^2 : Proof
Introduction
The algebraic identity (a+b)^2 = a^2 + 2ab + b^2 is a fundamental concept in mathematics, widely used in various fields such as algebra, geometry, and calculus. In this article, we will provide a proof of this identity, which is essential for understanding many mathematical concepts.
Proof
To prove the identity, we can start by using the distributive property of multiplication over addition, which states that:
(a+b)^2 = (a+b) × (a+b)
Now, we can expand the right-hand side of the equation using the distributive property:
(a+b) × (a+b) = a × (a+b) + b × (a+b)
Next, we can expand each term further:
a × (a+b) = a^2 + ab b × (a+b) = ba + b^2
Now, we can combine these two expressions:
(a+b)^2 = a^2 + ab + ba + b^2
Simplifying the Expression
Notice that the two middle terms, ab and ba, are equal:
ab = ba
This is known as the commutative property of multiplication. Therefore, we can simplify the expression by combining these two terms:
(a+b)^2 = a^2 + 2ab + b^2
Thus, we have successfully proved the algebraic identity (a+b)^2 = a^2 + 2ab + b^2.
Conclusion
In conclusion, the proof of (a+b)^2 = a^2 + 2ab + b^2 is a straightforward application of the distributive property and the commutative property of multiplication. This identity is a fundamental tool in mathematics, and its proof provides a solid foundation for understanding various mathematical concepts.
