(a-3b)(a+3b)-(a-3b)^2

Jasonbradley

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Jun 16, 2024

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Simplifying the Expression: (a-3b)(a+3b)-(a-3b)^2

This article will guide you through simplifying the algebraic expression (a-3b)(a+3b)-(a-3b)^2. We will use the concepts of difference of squares and perfect square trinomials to reach the simplified form.

Understanding the Expression

Let's break down the expression:

• (a-3b)(a+3b) This is a product of two binomials, which resembles the difference of squares pattern.
• (a-3b)^2 This represents the square of a binomial, following the pattern of a perfect square trinomial.

Applying the Patterns

1. Difference of Squares: (a-3b)(a+3b) = a^2 - (3b)^2 = a^2 - 9b^2

2. Perfect Square Trinomial: (a-3b)^2 = a^2 - 2(a)(3b) + (3b)^2 = a^2 - 6ab + 9b^2

Combining the Results

Now, we can substitute the simplified expressions back into the original expression:

(a-3b)(a+3b)-(a-3b)^2 = (a^2 - 9b^2) - (a^2 - 6ab + 9b^2)

Simplifying Further

Finally, we distribute the negative sign and combine like terms:

a^2 - 9b^2 - a^2 + 6ab - 9b^2 = 6ab - 18b^2

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