(a-b)%3Da%5E2-b%5E2-formula.jpeg "(a+b)(a-b)=a^2-b^2 Formula")
The Difference of Squares Formula: (a+b)(a-b)=a^2-b^2
The difference of squares formula is a fundamental algebraic identity that helps simplify expressions and solve equations. It states that the product of the sum and difference of two terms is equal to the difference of their squares.
The formula:
(a + b)(a - b) = a² - b²
Understanding the Formula:
To understand how this formula works, we can expand the left-hand side using the distributive property:
(a + b)(a - b) = a(a - b) + b(a - b) = a² - ab + ab - b² = a² - b²
The middle terms, -ab and +ab, cancel each other out, leaving us with the difference of squares, a² - b².
Applications of the Difference of Squares Formula:
This formula has numerous applications in algebra, including:
- Factoring expressions: The formula can be used to factor expressions that are in the form of a² - b². For example, x² - 9 can be factored as (x + 3)(x - 3).
- Simplifying expressions: It can be used to simplify expressions that involve the product of a sum and difference. For example, (2x + 5)(2x - 5) can be simplified to 4x² - 25.
- Solving equations: The formula can be used to solve equations where one side is the difference of squares. For example, the equation x² - 16 = 0 can be solved using the difference of squares formula to get x = ±4.
Examples:
Here are some examples of how the difference of squares formula can be applied:
- Factoring: Factor the expression x² - 49.
- Using the formula, we can factor it as (x + 7)(x - 7).
- Simplifying: Simplify the expression (3x + 2y)(3x - 2y).
- Using the formula, we get 9x² - 4y².
- Solving: Solve the equation 4x² - 25 = 0.
- Using the formula, we get (2x + 5)(2x - 5) = 0.
- This gives us two solutions: x = -5/2 and x = 5/2.
In Conclusion:
The difference of squares formula is a valuable tool in algebra, enabling us to factor expressions, simplify equations, and solve for unknown variables. By understanding and applying this formula, we can simplify complex mathematical problems and gain a deeper understanding of algebraic concepts.