%5E2.jpeg "(x-7)^2")
Understanding (x - 7)^2
The expression (x - 7)^2 is a simple but important example of a squared binomial. Understanding this expression is crucial for various mathematical concepts, including:
- Algebraic manipulation
- Quadratic equations
- Graphing functions
Let's break down what (x - 7)^2 represents and how to work with it:
Expanding the Expression
(x - 7)^2 means (x - 7) multiplied by itself:
(x - 7)^2 = (x - 7) * (x - 7)
To expand this, we use the FOIL method:
- First: x * x = x^2
- Outer: x * -7 = -7x
- Inner: -7 * x = -7x
- Last: -7 * -7 = 49
Combining like terms, we get:
(x - 7)^2 = x^2 - 7x - 7x + 49 = x^2 - 14x + 49
Key Properties
- Perfect Square Trinomial: (x - 7)^2 is a perfect square trinomial, meaning it is the result of squaring a binomial.
- Symmetry: Notice the coefficients of the terms. The coefficient of the middle term (-14x) is twice the product of the terms in the original binomial (x and -7).
- Constant Term: The constant term (49) is the square of the second term in the binomial (-7).
Applications
Understanding (x - 7)^2 has various applications:
- Solving Quadratic Equations: (x - 7)^2 = 0 can be solved by taking the square root of both sides, leading to x = 7.
- Graphing Parabolas: The expression represents a parabola shifted 7 units to the right.
- Factoring Expressions: Recognizing (x - 7)^2 allows for easier factoring of related expressions.
Conclusion
(x - 7)^2 is a fundamental expression in algebra with significant implications in various mathematical fields. Understanding its expansion, properties, and applications provides a solid foundation for further mathematical exploration.