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The Binomial Square Formula: (a+b)^2 = a^2 + 2ab + b^2
The binomial square formula, also known as the "(a+b)^2 formula", is a fundamental concept in algebra that describes the expansion of the square of a binomial expression. In this article, we will explore the formula, its proof, and some examples to illustrate its use.
What is the Binomial Square Formula?
The binomial square formula is a mathematical expression that describes the result of squaring a binomial expression, which is an expression consisting of two terms combined using addition or subtraction. The formula is:
(a+b)^2 = a^2 + 2ab + b^2
Where "a" and "b" are the two terms of the binomial expression.
Proof of the Binomial Square Formula
The proof of the binomial square formula is based on the distributive property of multiplication over addition. Let's start with the left-hand side of the equation:
(a+b)^2 = (a+b)(a+b)
Using the distributive property, we can expand the right-hand side as follows:
(a+b)(a+b) = a(a+b) + b(a+b)
= a^2 + ab + ba + b^2
Now, let's simplify the expression by combining like terms:
= a^2 + 2ab + b^2
Which is the right-hand side of the binomial square formula.
Examples of the Binomial Square Formula
Let's consider some examples to illustrate the use of the binomial square formula:
Example 1
Expand (x+3)^2 using the binomial square formula.
(x+3)^2 = x^2 + 2(x)(3) + 3^2
= x^2 + 6x + 9
Example 2
Expand (2y-4)^2 using the binomial square formula.
(2y-4)^2 = (2y)^2 + 2(2y)(-4) + (-4)^2
= 4y^2 - 16y + 16
Conclusion
The binomial square formula is a powerful tool for expanding the square of a binomial expression. It is a fundamental concept in algebra and is used extensively in various mathematical applications, including calculus, geometry, and trigonometry. By mastering the binomial square formula, you will be able to simplify complex expressions and solve a wide range of mathematical problems.
