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

3 min read Jun 16, 2024

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

The equation (a + b)^2 = a^2 + b^2 + 2ab is a fundamental algebraic identity that holds true for all values of a and b. This identity is often used to simplify expressions, solve equations, and understand the relationship between squares and products.

Here's a breakdown of the proof:

Expanding the Left-Hand Side

The left-hand side of the equation is (a + b)^2. This means we are squaring the binomial (a + b).

Recall that squaring a term means multiplying it by itself:

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

Applying the Distributive Property

Now, we use the distributive property to expand the product. The distributive property states that the product of a sum and a number is equal to the sum of the products of the number and each term in the sum.

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

Further Expansion

We distribute again to expand the right-hand side:

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

Simplifying

Finally, we simplify the expression by combining like terms and using the commutative property of multiplication (a * b = b * a):

a * a + a * b + b * a + b * b = a^2 + 2ab + b^2

Conclusion

We have shown that (a + b)^2 can be expanded and simplified to a^2 + b^2 + 2ab. This proves the identity, demonstrating that the two sides of the equation are equivalent for any values of a and b.

This proof highlights the power of algebraic manipulation and the importance of understanding fundamental identities in mathematics.