(x%2Bb)-formula-example.jpeg "(x+a)(x+b) Formula Example")
Understanding the (x + a)(x + b) Formula
The formula (x + a)(x + b) is a fundamental concept in algebra, allowing us to expand and simplify expressions involving binomials. This formula helps us understand how to multiply two binomials and derive the resulting quadratic expression.
The Formula
The formula states that:
(x + a)(x + b) = x² + (a + b)x + ab
Where:
- x is a variable
- a and b are constants
Explanation
The formula can be derived by using the distributive property of multiplication:
-
Expand the first term (x + a):
- x * (x + b) = x² + bx
-
Expand the second term (x + b):
- a * (x + b) = ax + ab
-
Combine the results:
- x² + bx + ax + ab
-
Simplify by grouping like terms:
- x² + (a + b)x + ab
Example
Let's illustrate the formula with an example:
Expand (x + 3)(x + 5)
Using the formula:
- a = 3
- b = 5
(x + 3)(x + 5) = x² + (3 + 5)x + 3 * 5
Simplifying:
(x + 3)(x + 5) = x² + 8x + 15
Applications
The (x + a)(x + b) formula is widely used in various mathematical contexts, including:
- Factoring quadratic expressions: We can use the formula to factor quadratic expressions by reversing the expansion process.
- Solving quadratic equations: By applying the formula, we can simplify quadratic equations and find their solutions.
- Algebraic manipulations: The formula helps in simplifying complex expressions involving binomials.
Conclusion
The (x + a)(x + b) formula is an essential tool in algebra for expanding and simplifying expressions involving binomials. Understanding this formula provides a foundation for tackling various algebraic problems and deepening your understanding of quadratic expressions.