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

3 min read Jun 17, 2024
(x-3)(x+3)-(x-3)^2

Simplifying the Expression (x-3)(x+3)-(x-3)^2

This article will guide you through simplifying the expression (x-3)(x+3)-(x-3)^2. We will utilize algebraic techniques and properties to arrive at the simplest form.

Understanding the Expression

The expression involves two key components:

  1. (x-3)(x+3): This is a product of two binomials, which resembles the form of the difference of squares factorization pattern: (a+b)(a-b) = a^2 - b^2.

  2. (x-3)^2: This is a squared binomial, which expands using the square of a binomial pattern: (a-b)^2 = a^2 - 2ab + b^2.

Simplifying the Expression Step-by-Step

  1. Expand the first term: Using the difference of squares pattern, we get: (x-3)(x+3) = x^2 - 3^2 = x^2 - 9

  2. Expand the second term: Using the square of a binomial pattern, we get: (x-3)^2 = x^2 - 2(x)(3) + 3^2 = x^2 - 6x + 9

  3. Combine the expanded terms: Substitute the expanded terms back into the original expression: (x-3)(x+3) - (x-3)^2 = (x^2 - 9) - (x^2 - 6x + 9)

  4. Distribute the negative sign: (x^2 - 9) - (x^2 - 6x + 9) = x^2 - 9 - x^2 + 6x - 9

  5. Combine like terms: Notice that x^2 and -x^2 cancel out. Combining the constants, we get: x^2 - 9 - x^2 + 6x - 9 = 6x - 18

Final Result

Therefore, the simplified form of the expression (x-3)(x+3)-(x-3)^2 is 6x - 18.

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