(x-1)^3-(x+3)(x^2-3x+9)+3x^2=25

Jasonbradley

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Jun 17, 2024

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Solving the Equation: (x-1)^3 - (x+3)(x^2 - 3x + 9) + 3x^2 = 25

This article will walk you through the steps of solving the equation (x-1)^3 - (x+3)(x^2 - 3x + 9) + 3x^2 = 25. We will utilize algebraic manipulations to simplify the equation and isolate the variable x.

Expanding and Simplifying

1. Expand the cubes:
   Begin by expanding the term (x-1)^3 using the binomial theorem or by multiplying (x-1) by itself three times.

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

2. Expand the product:
   The second term involves a product of two expressions. Notice that (x^2 - 3x + 9) is a pattern resembling the difference of cubes.

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

3. Substitute and simplify: Now, substitute the expanded terms back into the original equation.

   x^3 - 3x^2 + 3x - 1 - (x^3 + 27) + 3x^2 = 25

   Simplify by combining like terms.

   3x - 28 = 25

Isolating the Variable

1. Isolate x:
   Add 28 to both sides of the equation to move the constant term to the right side.

   3x = 53

2. Solve for x:
   Divide both sides of the equation by 3 to isolate x.

   x = 53/3

Solution

Therefore, the solution to the equation (x-1)^3 - (x+3)(x^2 - 3x + 9) + 3x^2 = 25 is x = 53/3.