%5E2.jpeg "(x-9)^2")
Understanding (x-9)^2
(x-9)^2 is a mathematical expression that represents the square of the binomial (x-9). To understand this expression better, let's break down its components and explore its properties:
Binomial Expansion
The expression (x-9)^2 is essentially the product of (x-9) with itself:
(x-9)^2 = (x-9) * (x-9)
To expand this, we can use the FOIL method (First, Outer, Inner, Last):
- First: x * x = x^2
- Outer: x * -9 = -9x
- Inner: -9 * x = -9x
- Last: -9 * -9 = 81
Adding these terms together, we get the expanded form:
(x-9)^2 = x^2 - 9x - 9x + 81 = x^2 - 18x + 81
Properties of the Expression
- Quadratic Equation: The expanded form of (x-9)^2 is a quadratic equation, which is a polynomial equation of the second degree.
- Vertex Form: The expression can be rewritten in vertex form, which is a standard form for quadratic equations that reveals the vertex of the parabola:
(x-9)^2 = (x-9)^2 + 0
- Vertex: The vertex of the parabola represented by the expression is at the point (9, 0).
Applications
The expression (x-9)^2 has various applications in mathematics and other fields, including:
- Algebra: Solving quadratic equations and simplifying expressions involving binomials.
- Calculus: Finding derivatives and integrals of functions involving binomials.
- Geometry: Calculating areas and volumes of shapes derived from the expression.
- Physics: Modeling physical phenomena that can be represented by quadratic functions.
Understanding the properties and expansion of (x-9)^2 is crucial for working with quadratic equations and various mathematical concepts.