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(2x-3)(2x+3)

2 min read Jun 16, 2024
(2x-3)(2x+3)

Expanding (2x-3)(2x+3)

In algebra, expanding expressions often involves applying the distributive property. This property allows us to multiply terms within parentheses by a factor outside. When we encounter expressions like (2x-3)(2x+3), we can use a handy pattern called the "difference of squares" to simplify the expansion.

Understanding the Difference of Squares Pattern

The difference of squares pattern states:

** (a - b)(a + b) = a² - b² **

This pattern arises because when we expand the left side, the middle terms cancel out:

  • (a - b)(a + b) = a(a + b) - b(a + b)
  • = a² + ab - ba - b²
  • = a² - b²

Applying the Pattern to (2x-3)(2x+3)

  1. Identify 'a' and 'b':

    • In this case, a = 2x and b = 3

  2. Apply the pattern:

    • (2x - 3)(2x + 3) = (2x)² - (3)²

  3. Simplify:

    • = 4x² - 9

Therefore, the expanded form of (2x-3)(2x+3) is 4x² - 9.

Using the FOIL Method (Alternative Approach)

While the difference of squares pattern provides a quicker solution, we can also use the FOIL method (First, Outer, Inner, Last) to expand the expression:

  • First: 2x * 2x = 4x²
  • Outer: 2x * 3 = 6x
  • Inner: -3 * 2x = -6x
  • Last: -3 * 3 = -9

Combining like terms, we get 4x² + 6x - 6x - 9, which simplifies to 4x² - 9.

As you can see, both methods lead to the same answer. Choosing the most efficient approach depends on your comfort level and the specific problem you are tackling.

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