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(x-1)^3+(2-x)(4+2x+x^2)+3x(x+2)=16

2 min read Jun 17, 2024
(x-1)^3+(2-x)(4+2x+x^2)+3x(x+2)=16

Solving the Equation: (x-1)^3 + (2-x)(4+2x+x^2) + 3x(x+2) = 16

This article will walk through the steps to solve the given equation:

(x-1)^3 + (2-x)(4+2x+x^2) + 3x(x+2) = 16

Simplifying the Equation

  1. Expand the cubes and products:

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

  2. Substitute the expanded terms back into the equation: x^3 - 3x^2 + 3x - 1 + 8 - x^3 + 3x^2 + 6x = 16

  3. Combine like terms: 9x + 7 = 16

Solving for x

  1. Isolate the x term: 9x = 9

  2. Solve for x by dividing both sides by 9: x = 1

Solution

Therefore, the solution to the equation (x-1)^3 + (2-x)(4+2x+x^2) + 3x(x+2) = 16 is x = 1.

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