(a+b)^2=a^2+b^2+2ab Proof

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Jul 03, 2024

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Proof of (a+b)^2 = a^2 + b^2 + 2ab

In mathematics, one of the most fundamental and widely used identities is the expansion of (a+b)^2. This identity is a crucial tool in various mathematical domains, including algebra, geometry, and calculus. In this article, we will provide a step-by-step proof of the identity (a+b)^2 = a^2 + b^2 + 2ab.

Proof

To prove this identity, we will start by using the distributive property of multiplication over addition. This property states that for any numbers a, b, and c, the following equation holds:

a(b+c) = ab + ac

Now, let's consider the expression (a+b)^2. We can rewrite this expression as:

(a+b)^2 = (a+b)(a+b)

Using the distributive property, we can expand the right-hand side of the equation as:

(a+b)(a+b) = a(a+b) + b(a+b)

= a^2 + ab + ba + b^2

Next, we can combine like terms:

a^2 + ab + ba + b^2 = a^2 + 2ab + b^2

Thus, we have proved that:

(a+b)^2 = a^2 + 2ab + b^2

This identity is a fundamental tool in mathematics and is widely used in various mathematical applications.

Conclusion

In conclusion, we have provided a step-by-step proof of the identity (a+b)^2 = a^2 + 2ab + b^2. This identity is a crucial tool in mathematics and has numerous applications in algebra, geometry, and calculus. By using the distributive property of multiplication over addition, we have successfully expanded the expression (a+b)^2 and derived the desired result.