(x-10)-identity.jpeg "(x+8)(x-10) Identity")
Understanding the (x+8)(x-10) Identity
The expression (x+8)(x-10) is a simple example of a quadratic expression that can be expanded using the distributive property or the FOIL method. Let's break down how to expand this identity and the key concepts involved.
Expanding the Identity
1. Distributive Property:
This method involves multiplying each term in the first set of parentheses by each term in the second set of parentheses.
(x+8)(x-10) = x(x-10) + 8(x-10)
2. FOIL Method:
This is a mnemonic for remembering the order of multiplication: First, Outer, Inner, Last.
- First: Multiply the first terms of each binomial: x * x = x²
- Outer: Multiply the outer terms of the binomials: x * -10 = -10x
- Inner: Multiply the inner terms of the binomials: 8 * x = 8x
- Last: Multiply the last terms of each binomial: 8 * -10 = -80
Adding all these terms together, we get: x² - 10x + 8x - 80
Simplifying the Expression
Combining the like terms (-10x + 8x), we arrive at the simplified form:
(x+8)(x-10) = x² - 2x - 80
Key Concepts
- Quadratic Expression: A polynomial expression with the highest power of the variable being 2.
- Distributive Property: Allows multiplication of a sum by a number.
- FOIL Method: A specific way to apply the distributive property to binomial multiplication.
Applications
The expanded form of this identity is useful in various mathematical contexts:
- Solving Quadratic Equations: By setting the expression equal to zero, we can solve for the roots of the equation.
- Factoring Quadratic Expressions: This identity helps us factor quadratic expressions into their binomial components.
- Graphing Parabolas: Understanding the expanded form helps in determining the vertex, intercepts, and general shape of the parabola represented by the equation.
By understanding how to expand and simplify the (x+8)(x-10) identity, we gain valuable tools for working with quadratic expressions and their applications in algebra, calculus, and other areas of mathematics.