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(2x+3)^2+(2x+3)(2x-6)+(x-3)^2

3 min read Jun 16, 2024
(2x+3)^2+(2x+3)(2x-6)+(x-3)^2

Simplifying the Expression: (2x+3)^2+(2x+3)(2x-6)+(x-3)^2

This article will guide you through the process of simplifying the given algebraic expression: (2x+3)^2+(2x+3)(2x-6)+(x-3)^2.

Understanding the Components

Before we start simplifying, let's break down the expression:

  • (2x+3)^2: This represents the square of the binomial (2x+3), which means multiplying it by itself.
  • (2x+3)(2x-6): This is the product of two binomials, (2x+3) and (2x-6).
  • (x-3)^2: Similar to the first term, this represents the square of the binomial (x-3).

Simplifying Using the Distributive Property and FOIL

We can simplify the expression using the distributive property and the FOIL method:

  1. Expanding (2x+3)^2: (2x+3)^2 = (2x+3)(2x+3) Using FOIL: = (2x * 2x) + (2x * 3) + (3 * 2x) + (3 * 3) = 4x^2 + 6x + 6x + 9 = 4x^2 + 12x + 9

  2. Expanding (2x+3)(2x-6): Using FOIL: = (2x * 2x) + (2x * -6) + (3 * 2x) + (3 * -6) = 4x^2 - 12x + 6x - 18 = 4x^2 - 6x - 18

  3. Expanding (x-3)^2: (x-3)^2 = (x-3)(x-3) Using FOIL: = (x * x) + (x * -3) + (-3 * x) + (-3 * -3) = x^2 - 6x + 9

Combining the Simplified Terms

Now, we have the simplified forms of each term:

  • 4x^2 + 12x + 9
  • 4x^2 - 6x - 18
  • x^2 - 6x + 9

Let's combine them:

(4x^2 + 12x + 9) + (4x^2 - 6x - 18) + (x^2 - 6x + 9)

Combining like terms:

= (4x^2 + 4x^2 + x^2) + (12x - 6x - 6x) + (9 - 18 + 9) = 9x^2 + 0x

Final Simplified Expression

The simplified form of the expression (2x+3)^2+(2x+3)(2x-6)+(x-3)^2 is 9x^2.

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