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(x-1)^3-(x+3)(x^2-3x+9)+3(x^2-4)=2

3 min read Jun 17, 2024
(x-1)^3-(x+3)(x^2-3x+9)+3(x^2-4)=2

Solving the Equation: (x-1)^3-(x+3)(x^2-3x+9)+3(x^2-4)=2

This article will guide you through the process of solving the equation (x-1)^3-(x+3)(x^2-3x+9)+3(x^2-4)=2. We will break down the problem step-by-step, simplifying the expression and arriving at the solution.

Expanding and Simplifying

  1. Expand the cubes: We begin by expanding the cube term using the formula (a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3.

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

  2. Expand the product: The expression (x+3)(x^2-3x+9) is a product of the sum and difference of two cubes. We can use the formula (a+b)(a^2-ab+b^2) = a^3 + b^3 to simplify.

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

  3. Simplify the expression: Substitute the expanded forms back into the original equation:

    (x^3 - 3x^2 + 3x - 1) - (x^3 + 27) + 3(x^2 - 4) = 2

  4. Combine like terms: Remove the parentheses and combine the terms with similar powers of x.

    x^3 - 3x^2 + 3x - 1 - x^3 - 27 + 3x^2 - 12 = 2

    3x - 40 = 2

Solving for x

  1. Isolate the x term: Add 40 to both sides of the equation.

    3x = 42

  2. Solve for x: Divide both sides by 3.

    x = 14

Conclusion

Therefore, the solution to the equation (x-1)^3-(x+3)(x^2-3x+9)+3(x^2-4)=2 is x = 14.

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