%5E2-formula.jpeg "(ax+b)^2 Formula")
The Formula for (ax + b)^2
The formula for (ax + b)^2 is a fundamental concept in algebra and is widely used in various mathematical operations. In this article, we will explore the formula, its derivation, and some examples to illustrate its application.
The Formula
The formula for (ax + b)^2 is:
(ax + b)^2 = a^2x^2 + 2abx + b^2
This formula is used to expand the square of a binomial expression, which is an expression consisting of two terms.
Derivation of the Formula
The formula for (ax + b)^2 can be derived by using the distributive property of multiplication over addition. Let's start by multiplying (ax + b) by itself:
(ax + b)(ax + b) = (ax + b)ax + (ax + b)b
Using the distributive property, we can expand the right-hand side of the equation as:
(ax + b)ax + (ax + b)b = a^2x^2 + abx + abx + b^2
Combine like terms:
a^2x^2 + 2abx + b^2
Thus, we have derived the formula for (ax + b)^2.
Examples
Example 1
Find the expansion of (2x + 3)^2.
Using the formula, we get:
(2x + 3)^2 = 2^2x^2 + 2(2)(3)x + 3^2 = 4x^2 + 12x + 9
Example 2
Find the expansion of (x + 5)^2.
Using the formula, we get:
(x + 5)^2 = x^2 + 2(1)(5)x + 5^2 = x^2 + 10x + 25
Example 3
Find the expansion of (3x - 2)^2.
Using the formula, we get:
(3x - 2)^2 = 3^2x^2 + 2(3)(-2)x + (-2)^2 = 9x^2 - 12x + 4
Conclusion
In conclusion, the formula for (ax + b)^2 is a powerful tool for expanding the square of a binomial expression. By understanding and applying this formula, you can simplify complex algebraic expressions and solve a wide range of mathematical problems.
