%5E2%3Dx%5E2%2B2x%2B1.jpeg "(x+1)^2=x^2+2x+1")
Understanding the Equation (x + 1)² = x² + 2x + 1
This equation represents a fundamental concept in algebra: the square of a binomial. Let's break it down to understand its meaning and how it works.
The Left Hand Side: (x + 1)²
- (x + 1)² means (x + 1) multiplied by itself. We can write this out as:
- (x + 1) * (x + 1)
The Right Hand Side: x² + 2x + 1
This expression represents the expanded form of the left-hand side. It's obtained by applying the distributive property of multiplication:
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Multiply the first term of the first binomial by each term in the second binomial:
- x * x = x²
- x * 1 = x
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Multiply the second term of the first binomial by each term in the second binomial:
- 1 * x = x
- 1 * 1 = 1
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Add all the resulting terms:
- x² + x + x + 1 = x² + 2x + 1
Why is this important?
This equation demonstrates how to expand a binomial squared. This pattern is crucial for:
- Simplifying algebraic expressions: When you encounter (x + 1)² in an expression, you can replace it with x² + 2x + 1, making the expression simpler to work with.
- Solving equations: Recognizing this pattern helps in factoring quadratic equations and finding their solutions.
- Understanding the relationship between algebraic expressions and their geometric representations: The equation is linked to the area of a square with sides of length (x + 1).
Generalizing the Pattern
The pattern observed in this equation applies to any binomial squared:
(a + b)² = a² + 2ab + b²
This is a useful formula to remember and apply in various mathematical contexts.