(x-3)^3-(x-3)(x^2+3x+9)+9(x+1)^2=15

Jasonbradley

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Jun 17, 2024

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Solving the Equation: (x-3)^3 - (x-3)(x^2+3x+9) + 9(x+1)^2 = 15

This equation presents a challenge due to its complex structure. Let's break it down step-by-step to find its solution.

Understanding the Equation

• Recognizing patterns: The expression (x^2 + 3x + 9) is a special pattern, it's the expansion of (x + 3)^2. This recognition is crucial to simplify the equation.

Simplifying the Equation

1. Expanding the cubes:

   • (x-3)^3 = (x-3)(x-3)(x-3) = x^3 - 9x^2 + 27x - 27
   • (x+1)^2 = (x+1)(x+1) = x^2 + 2x + 1

2. Applying the difference of cubes formula:

   • The expression (x-3)(x^2 + 3x + 9) represents the difference of cubes formula: a^3 - b^3 = (a-b)(a^2 + ab + b^2), where a = x and b = 3. Therefore, this expression simplifies to x^3 - 27.

3. Substituting and rearranging:

   • Now, substituting the expanded terms and simplifying, we get: x^3 - 9x^2 + 27x - 27 - (x^3 - 27) + 9(x^2 + 2x + 1) = 15 -9x^2 + 27x + 9x^2 + 18x + 9 = 15

4. Combining like terms:

   • 45x + 9 = 15
   • 45x = 6
   • x = 6/45

The Solution

Therefore, the solution to the equation (x-3)^3 - (x-3)(x^2+3x+9) + 9(x+1)^2 = 15 is x = 2/15.