%5E2.jpeg "(12-x)^2")
Understanding (12 - x)^2
The expression (12 - x)^2 represents the square of a binomial, which is an algebraic expression with two terms. Let's break down its meaning and how to expand it.
What Does Squaring a Binomial Mean?
Squaring a binomial means multiplying it by itself. In this case:
(12 - x)^2 = (12 - x) * (12 - x)
Expanding the Expression
To expand the expression, we can use the distributive property or the FOIL method:
1. Using the Distributive Property:
- First: Distribute the first term of the first binomial (12) to both terms of the second binomial:
- 12 * 12 = 144
- 12 * (-x) = -12x
- Second: Distribute the second term of the first binomial (-x) to both terms of the second binomial:
- (-x) * 12 = -12x
- (-x) * (-x) = x^2
- Combine the terms: 144 - 12x - 12x + x^2
2. Using the FOIL Method:
- F: First terms: 12 * 12 = 144
- O: Outer terms: 12 * (-x) = -12x
- I: Inner terms: (-x) * 12 = -12x
- L: Last terms: (-x) * (-x) = x^2
- Combine the terms: 144 - 12x - 12x + x^2
Simplified Expression
After combining like terms, the expanded form of (12 - x)^2 is:
x^2 - 24x + 144
Applications
Understanding how to expand binomials like (12 - x)^2 is crucial in various areas of mathematics, including:
- Algebraic manipulation: Simplifying expressions, solving equations
- Calculus: Finding derivatives and integrals
- Statistics and Probability: Working with probability distributions
By learning this concept, you gain a valuable tool for tackling more complex mathematical problems.