%5E2.jpeg "(x+7)^2")
Understanding (x+7)^2
The expression (x+7)^2 represents the square of the binomial (x+7). This means we're multiplying the binomial by itself. Let's break down how to expand this expression and understand its significance.
Expanding the Expression
To expand (x+7)^2, we can apply the distributive property, also known as the FOIL method (First, Outer, Inner, Last):
- First: x * x = x^2
- Outer: x * 7 = 7x
- Inner: 7 * x = 7x
- Last: 7 * 7 = 49
Adding all the terms together, we get:
(x+7)^2 = x^2 + 7x + 7x + 49
Simplifying the expression, we arrive at:
(x+7)^2 = x^2 + 14x + 49
Understanding the Result
The expanded form of (x+7)^2 reveals several key concepts:
- Perfect Square Trinomial: The expression x^2 + 14x + 49 is a perfect square trinomial, meaning it results from squaring a binomial.
- Coefficient of the Linear Term: The coefficient of the linear term (14x) is twice the product of the terms in the original binomial (2 * 7 * x = 14x).
- Constant Term: The constant term (49) is the square of the second term in the original binomial (7^2 = 49).
Applications
Understanding (x+7)^2 and its expansion has various applications in mathematics, including:
- Algebraic Manipulation: This expansion is helpful for simplifying expressions, solving equations, and performing other algebraic operations.
- Quadratic Equations: The expanded form can be used to identify and factor quadratic equations, making it easier to solve for their roots.
- Calculus: The concept of expanding binomials is fundamental in calculus for understanding derivatives and integrals.
In conclusion, understanding the expansion of (x+7)^2 is crucial for mastering algebraic concepts and solving various mathematical problems. By applying the distributive property and recognizing the characteristics of a perfect square trinomial, we can effectively expand and interpret this expression.