(x-2)^3-(x+5)(x^2-5x+25)+6x^2=11

Jasonbradley

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Jun 17, 2024

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Solving the Equation: (x-2)^3 - (x+5)(x^2-5x+25) + 6x^2 = 11

This article will guide you through the steps to solve the equation: (x-2)^3 - (x+5)(x^2-5x+25) + 6x^2 = 11.

Understanding the Equation

The equation involves several algebraic operations, including:

• Cubing a binomial: (x-2)^3
• Expanding a product of a binomial and a trinomial: (x+5)(x^2-5x+25)
• Combining like terms: 6x^2

To solve for x, we need to simplify the equation and isolate x.

Step-by-Step Solution

1. Expand the cube: (x-2)^3 = (x-2)(x-2)(x-2) = x^3 - 6x^2 + 12x - 8

2. Expand the product using the difference of cubes pattern: (x+5)(x^2-5x+25) = x^3 + 5^3 = x^3 + 125

3. Substitute the expanded terms back into the original equation: (x^3 - 6x^2 + 12x - 8) - (x^3 + 125) + 6x^2 = 11

4. Simplify the equation by combining like terms: -6x^2 + 12x - 8 - 125 + 6x^2 = 11 12x - 133 = 11

5. Isolate x by adding 133 to both sides: 12x = 144

6. Solve for x by dividing both sides by 12: x = 12

Conclusion

Therefore, the solution to the equation (x-2)^3 - (x+5)(x^2-5x+25) + 6x^2 = 11 is x = 12.