(x%2Bb)-formula-example.jpeg "(x+a)(x+b) Formula Example")
The Formula for (x+a)(x+b)
The formula for multiplying two binomials, (x+a) and (x+b), is a fundamental concept in algebra. It is widely used in various mathematical operations, such as factoring, simplifying expressions, and solving equations. In this article, we will explore the formula, its proof, and provide examples to illustrate its application.
The Formula
The formula for multiplying (x+a) and (x+b) is:
(x+a)(x+b) = x^2 + (a+b)x + ab
This formula states that when you multiply two binomials, (x+a) and (x+b), the result is a quadratic expression consisting of three terms: a squared term (x^2), a linear term ((a+b)x), and a constant term (ab).
Proof
To prove this formula, let's expand the product of (x+a) and (x+b) using the distributive property of multiplication over addition:
(x+a)(x+b) = x(x+b) + a(x+b) = x^2 + xb + ax + ab
Now, combine like terms:
= x^2 + (b+a)x + ab
Therefore, we have derived the formula: (x+a)(x+b) = x^2 + (a+b)x + ab.
Examples
Example 1: Expand (x+2)(x+3)
Using the formula, we get:
(x+2)(x+3) = x^2 + (2+3)x + (2)(3) = x^2 + 5x + 6
Example 2: Expand (x-4)(x+5)
Using the formula, we get:
(x-4)(x+5) = x^2 + (-4+5)x + (-4)(5) = x^2 + x - 20
Example 3: Simplify (x+1)(x+2)(x+3)
Using the formula repeatedly, we get:
(x+1)(x+2) = x^2 + 3x + 2 (x+1)(x^2 + 3x + 2) = x^3 + 4x^2 + 5x + 2 (x^3 + 4x^2 + 5x + 2)(x+3) = x^4 + 7x^3 + 17x^2 + 20x + 6
Therefore, (x+1)(x+2)(x+3) = x^4 + 7x^3 + 17x^2 + 20x + 6.
Conclusion
In conclusion, the formula for multiplying (x+a) and (x+b) is a powerful tool in algebra. It can be used to expand and simplify expressions, as well as solve equations. By mastering this formula, you will gain a better understanding of algebra and be able to tackle more complex mathematical problems.
