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(x+3)^3-x(3x+1)^2+(2x+1)(4x^2-2x+1)-3x^2=54

3 min read Jun 16, 2024
(x+3)^3-x(3x+1)^2+(2x+1)(4x^2-2x+1)-3x^2=54

Solving the Equation (x+3)^3 - x(3x+1)^2 + (2x+1)(4x^2-2x+1) - 3x^2 = 54

This article will guide you through the process of solving the given equation. We will simplify the equation step-by-step, employing algebraic techniques to arrive at the solution.

Expanding and Simplifying the Equation

  1. Expand the terms: Begin by expanding the powers and products in the equation:

    • (x+3)^3 = x^3 + 9x^2 + 27x + 27
    • x(3x+1)^2 = x(9x^2 + 6x + 1) = 9x^3 + 6x^2 + x
    • (2x+1)(4x^2-2x+1) = 8x^3 - 4x^2 + 2x + 4x^2 - 2x + 1 = 8x^3 + 1

  2. Substitute the expanded terms: Now substitute these expanded terms back into the original equation:

    x^3 + 9x^2 + 27x + 27 - (9x^3 + 6x^2 + x) + 8x^3 + 1 - 3x^2 = 54

  3. Simplify by combining like terms: Combine the terms with the same powers of x:

    (x^3 - 9x^3 + 8x^3) + (9x^2 - 6x^2 - 3x^2) + (27x - x) + (27 + 1) = 54

    This simplifies to: 0 + 0 + 26x + 28 = 54

Solving for x

  1. Isolate the variable: Move the constant term to the right side of the equation:

    26x = 54 - 28 26x = 26

  2. Solve for x: Divide both sides by 26 to find the value of x:

    x = 26 / 26 x = 1

Conclusion

Therefore, the solution to the equation (x+3)^3 - x(3x+1)^2 + (2x+1)(4x^2-2x+1) - 3x^2 = 54 is x = 1.

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