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

3 min read Jun 16, 2024
(x+3)(x^2-3x+9)-x(x-1)(x+1)=27

Solving the Equation (x+3)(x^2-3x+9)-x(x-1)(x+1)=27

This article will guide you through the steps to solve the equation (x+3)(x^2-3x+9)-x(x-1)(x+1)=27.

Understanding the Equation

The equation involves the product of two expressions:

  • (x+3)(x^2-3x+9) This is a special product known as the sum of cubes. It can be simplified using the formula: (a + b)(a^2 - ab + b^2) = a^3 + b^3.
  • x(x-1)(x+1) This is the product of three consecutive integers. It can be simplified using the formula: x(x-1)(x+1) = x^3 - x.

Step-by-Step Solution

  1. Expand the products:
    • Using the sum of cubes formula, we get: (x+3)(x^2-3x+9) = x^3 + 3^3 = x^3 + 27
    • Using the formula for the product of three consecutive integers, we get: x(x-1)(x+1) = x^3 - x
  2. Substitute the expanded expressions into the original equation:
    • (x^3 + 27) - (x^3 - x) = 27
  3. Simplify the equation:
    • x^3 + 27 - x^3 + x = 27
    • x + 27 = 27
  4. Isolate x:
    • x = 27 - 27
    • x = 0

Solution

Therefore, the solution to the equation (x+3)(x^2-3x+9)-x(x-1)(x+1)=27 is x = 0.

Verification

To verify the solution, substitute x = 0 back into the original equation:

  • (0+3)(0^2-3(0)+9) - 0(0-1)(0+1) = 27
  • (3)(9) - 0 = 27
  • 27 = 27

The equation holds true, confirming that x = 0 is indeed the solution.

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