%5E2%2Bb-formula.jpeg "(x+a)^2+b Formula")
The Formula for (x+a)^2+b: A Comprehensive Guide
Introduction
In algebra, expanding binomials is a crucial skill that helps in simplifying complex expressions. One of the most common binomials is (x+a)^2, which is a quadratic expression. In this article, we will explore the formula for (x+a)^2+b, where b is a constant. We will also delve into the expansion of the binomial and provide examples to illustrate the concept.
The Formula
The formula for (x+a)^2+b is:
(x+a)^2+b = x^2 + 2ax + a^2 + b
This formula is derived by expanding the binomial (x+a)^2 using the distributive property of multiplication over addition.
Expansion of the Binomial
To expand the binomial (x+a)^2, we need to multiply the binomial by itself:
(x+a)^2 = (x+a)(x+a)
Using the distributive property, we get:
(x+a)(x+a) = x^2 + ax + ax + a^2
Combine like terms:
(x+a)(x+a) = x^2 + 2ax + a^2
Now, add b to both sides of the equation:
(x+a)^2 + b = x^2 + 2ax + a^2 + b
Examples
Let's consider a few examples to illustrate the formula:
Example 1
Simplify the expression (x+2)^2 + 3:
(x+2)^2 + 3 = x^2 + 2(2)x + 2^2 + 3 (x+2)^2 + 3 = x^2 + 4x + 4 + 3 (x+2)^2 + 3 = x^2 + 4x + 7
Example 2
Expand the expression (x-3)^2 - 2:
(x-3)^2 - 2 = x^2 + 2(-3)x + (-3)^2 - 2 (x-3)^2 - 2 = x^2 - 6x + 9 - 2 (x-3)^2 - 2 = x^2 - 6x + 7
Conclusion
In this article, we have derived the formula for (x+a)^2+b and provided examples to illustrate the concept. This formula is essential in algebra and is used in various mathematical applications. By mastering this formula, you will be able to simplify complex expressions and solve quadratic equations with ease.
