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Solving the Equation: (x-1)^3+3(x-3)^2-(x+2)(x^2-2x+4)=(x+2)^3-(x-3)(x^2+9)-6x^2+5
This article will walk through the steps to solve the equation:
(x-1)^3+3(x-3)^2-(x+2)(x^2-2x+4)=(x+2)^3-(x-3)(x^2+9)-6x^2+5
Let's begin by expanding both sides of the equation.
Expanding the Left Side
(x-1)^3+3(x-3)^2-(x+2)(x^2-2x+4)
- Expanding (x-1)^3: Using the formula (a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3, we get: (x-1)^3 = x^3 - 3x^2 + 3x - 1
- Expanding 3(x-3)^2: Using the formula (a-b)^2 = a^2 - 2ab + b^2, we get: 3(x-3)^2 = 3(x^2 - 6x + 9) = 3x^2 - 18x + 27
- Expanding -(x+2)(x^2-2x+4): Using the formula (a+b)(a^2 - ab + b^2) = a^3 + b^3, we get: -(x+2)(x^2 - 2x + 4) = -(x^3 + 8) = -x^3 - 8
Combining the expanded terms, the left side becomes:
x^3 - 3x^2 + 3x - 1 + 3x^2 - 18x + 27 - x^3 - 8 = -15x + 18
Expanding the Right Side
(x+2)^3-(x-3)(x^2+9)-6x^2+5
- Expanding (x+2)^3: Using the formula (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3, we get: (x+2)^3 = x^3 + 6x^2 + 12x + 8
- Expanding -(x-3)(x^2+9): Using the formula (a-b)(a^2 + ab + b^2) = a^3 - b^3, we get: -(x-3)(x^2 + 9) = -(x^3 - 27) = -x^3 + 27
Combining the expanded terms, the right side becomes:
x^3 + 6x^2 + 12x + 8 - x^3 + 27 - 6x^2 + 5 = 12x + 40
Solving the Equation
Now we have:
-15x + 18 = 12x + 40
- Combining like terms: -27x = 22
- Solving for x: x = -22/27
Therefore, the solution to the equation is x = -22/27.