(a-b)%3Da%5E2-b%5E2-examples.jpeg "(a+b)(a-b)=a^2-b^2 Examples")
Algebraic Identity: (a+b)(a-b)=a^2-b^2 with Examples
In algebra, there are several important identities that help simplify complex expressions. One of these identities is the difference of squares formula, which states that (a+b)(a-b)=a^2-b^2. This identity is widely used in various mathematical operations, including simplifying expressions, solving equations, and factorizing polynomials.
Understanding the Formula
The formula (a+b)(a-b)=a^2-b^2 can be derived by multiplying the two binomials using the distributive property of multiplication over addition.
(a+b)(a-b) = a(a-b) + b(a-b) = a^2 - ab + ab - b^2 = a^2 - b^2
As you can see, the ab terms cancel out, leaving us with a^2 - b^2.
Examples
Let's see how this identity can be applied to various examples:
Example 1: Simplify the expression (x+3)(x-3)
Using the difference of squares formula, we get:
(x+3)(x-3) = x^2 - 3^2 = x^2 - 9
Example 2: Factorize the expression x^2 - 16
Using the difference of squares formula, we get:
x^2 - 16 = (x+4)(x-4)
Example 3: Simplify the expression (2a+5)(2a-5)
Using the difference of squares formula, we get:
(2a+5)(2a-5) = (2a)^2 - 5^2 = 4a^2 - 25
Example 4: Solve the equation (x+2)(x-2) = 9
Using the difference of squares formula, we get:
(x+2)(x-2) = x^2 - 2^2 = x^2 - 4 = 9
Now, we can solve for x:
x^2 = 13 x = ±√13
Conclusion
The identity (a+b)(a-b)=a^2-b^2 is a powerful tool in algebra, allowing us to simplify complex expressions, factorize polynomials, and solve equations. By applying this formula, we can simplify our work and make algebraic manipulations more efficient.
