(12-x)^2

2 min read Jun 16, 2024
(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.

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