-(a-b)-formula-examples.jpeg "(a+b) (a-b) Formula Examples")
(a+b)(a-b) Formula: Examples and Explanation
The (a+b)(a-b) formula is a fundamental concept in algebra that allows us to simplify complex expressions and solve equations. In this article, we will explore the formula, its derivation, and provide examples to illustrate its application.
What is the (a+b)(a-b) Formula?
The (a+b)(a-b) formula is a mathematical expression that represents the product of two binomials:
(a+b)(a-b) = a^2 - b^2
This formula is commonly used to simplify expressions and solve equations that involve the sum and difference of two terms.
Derivation of the Formula
To derive the formula, let's start with the product of two binomials:
(a+b)(a-b) = ?
Using the distributive property of multiplication over addition, we can expand the product as follows:
(a+b)(a-b) = a(a-b) + b(a-b)
Now, let's simplify each term:
a(a-b) = a^2 - ab
b(a-b) = ab - b^2
Combine the two terms:
(a+b)(a-b) = a^2 - ab + ab - b^2
Simplify the expression by combining like terms:
(a+b)(a-b) = a^2 - b^2
Thus, we have derived the (a+b)(a-b) formula.
Examples
Example 1: Simplify the Expression
Simplify the expression:
(x+3)(x-3)
Using the (a+b)(a-b) formula, we can write:
(x+3)(x-3) = x^2 - 3^2
= x^2 - 9
Example 2: Solve the Equation
Solve the equation:
(x+2)(x-2) = 0
Using the (a+b)(a-b) formula, we can rewrite the equation as:
x^2 - 2^2 = 0
x^2 - 4 = 0
Factor the quadratic expression:
(x-2)(x+2) = 0
Solve for x:
x - 2 = 0 or x + 2 = 0
x = 2 or x = -2
Example 3: Expand the Product
Expand the product:
(a+5)(a-5)
Using the (a+b)(a-b) formula, we can write:
(a+5)(a-5) = a^2 - 5^2
= a^2 - 25
In this example, we expanded the product using the formula.
Conclusion
The (a+b)(a-b) formula is a powerful tool for simplifying complex expressions and solving equations. By understanding the derivation and application of this formula, you can tackle a wide range of algebra problems with confidence.
