(x-b)-formula.jpeg "(x-a)(x-b) Formula")
The (x-a)(x-b) Formula: A Powerful Tool for Factoring Quadratic Expressions
Introduction
The (x-a)(x-b) formula is a widely used mathematical formula in algebra that helps to factorize quadratic expressions of the form ax^2 + bx + c. This formula is a powerful tool for solving quadratic equations and has numerous applications in various fields such as physics, engineering, and mathematics.
The Formula
The (x-a)(x-b) formula is given by:
x^2 - (a+b)x + ab = (x-a)(x-b)
where a and b are constants.
How it Works
To understand how the formula works, let's break it down:
- x^2 represents the square of the variable x.
- -(a+b)x represents the product of the sum of a and b, and the variable x.
- ab represents the product of a and b.
- (x-a)(x-b) represents the product of two binomials: (x-a) and (x-b).
When we multiply the two binomials, we get:
(x-a)(x-b) = x^2 - ax - bx + ab
Simplifying the expression, we get:
x^2 - (a+b)x + ab = (x-a)(x-b)
Examples
Let's look at a few examples to illustrate how the formula works:
Example 1
Factorize the quadratic expression: x^2 + 5x + 6
Using the formula, we get:
x^2 + 5x + 6 = (x+2)(x+3)
Example 2
Factorize the quadratic expression: x^2 - 3x - 4
Using the formula, we get:
x^2 - 3x - 4 = (x-4)(x+1)
Applications
The (x-a)(x-b) formula has numerous applications in various fields such as:
- Physics: In solving problems related to motion, force, and energy.
- Engineering: In designing electronic circuits, bridges, and buildings.
- Mathematics: In solving quadratic equations, graphing functions, and finding eigenvalues.
Conclusion
The (x-a)(x-b) formula is a powerful tool for factoring quadratic expressions and has wide-ranging applications in various fields. By mastering this formula, you can solve complex quadratic equations and unlock the secrets of algebra.
