%5E2-%3D-a%5E2-%2B-b%5E2-%2B-2ab.jpeg "(a+b)^2 = A^2 + B^2 + 2ab")
Understanding the Formula: (a + b)² = a² + b² + 2ab
The formula (a + b)² = a² + b² + 2ab is a fundamental concept in algebra, often referred to as the square of a binomial. It is used to simplify expressions involving the square of a sum. This article will explain the formula, its derivation, and provide practical examples to illustrate its application.
Deriving the Formula
The formula can be derived by expanding the expression (a + b)² using the distributive property of multiplication:
(a + b)² = (a + b)(a + b)
Expanding this product, we get:
(a + b)(a + b) = a(a + b) + b(a + b)
Applying the distributive property again, we obtain:
a(a + b) + b(a + b) = a² + ab + ba + b²
Combining the like terms, we arrive at the final formula:
a² + ab + ba + b² = a² + b² + 2ab
Therefore, (a + b)² = a² + b² + 2ab
Applying the Formula
The formula can be used to simplify various algebraic expressions. Here are a couple of examples:
Example 1: Simplify the expression (x + 3)²
Using the formula, we can directly substitute:
(x + 3)² = x² + 2(x)(3) + 3² = x² + 6x + 9
Example 2: Solve the equation (2x + 5)² = 49
Expanding the left side of the equation using the formula, we get:
(2x + 5)² = (2x)² + 2(2x)(5) + 5² = 4x² + 20x + 25
Now we have:
4x² + 20x + 25 = 49
Subtracting 49 from both sides:
4x² + 20x - 24 = 0
Simplifying by dividing by 4:
x² + 5x - 6 = 0
Factoring the quadratic equation:
(x + 6)(x - 1) = 0
Therefore, x = -6 or x = 1
Conclusion
The formula (a + b)² = a² + b² + 2ab is a valuable tool for simplifying algebraic expressions. It is crucial for understanding various concepts in algebra, particularly in solving quadratic equations and expanding expressions. By applying the formula correctly, you can efficiently simplify complex expressions and make calculations more manageable.