(x-b)-formula.jpeg "(x-a)(x-b) Formula")
The (x-a)(x-b) Formula: A Shortcut to Factoring Quadratics
The (x-a)(x-b) formula is a powerful tool for factoring quadratic expressions. It allows us to quickly and easily factor quadratic expressions in the form of ax² + bx + c, where a = 1.
Understanding the Formula
The formula states that:
(x - a)(x - b) = x² - (a + b)x + ab
Let's break down what each part represents:
- (x - a)(x - b): This is the factored form of the quadratic expression.
- x²: The first term in the expanded form represents the product of the 'x' terms in the factored form.
- -(a + b)x: The second term represents the sum of the products of the 'x' term from one factor and the constant term from the other.
- ab: The third term represents the product of the constant terms in the factored form.
How to Use the Formula
- Identify the coefficients: Start with a quadratic expression in the form x² + bx + c.
- Find two numbers: Find two numbers, a and b, that:
- Add up to b (the coefficient of the x term).
- Multiply to c (the constant term).
- Factor the quadratic: Once you've found a and b, you can write the factored form as (x - a)(x - b).
Example
Let's factor the quadratic expression x² - 5x + 6.
- Identify the coefficients: b = -5 and c = 6.
- Find two numbers: We need two numbers that add up to -5 and multiply to 6. These numbers are -2 and -3.
- Factor the quadratic: The factored form is (x - 2)(x - 3).
Advantages of Using the (x-a)(x-b) Formula
- Speed and Efficiency: This formula provides a quick and efficient way to factor quadratic expressions.
- Simplicity: It avoids the need for complex algebraic manipulations.
- Understanding the Relationships: It helps to visualize the relationship between the coefficients and the factors of the quadratic expression.
Conclusion
The (x-a)(x-b) formula is an invaluable tool for factoring quadratic expressions. By understanding this formula and its applications, you can confidently factor quadratic expressions and simplify algebraic problems.