(x+4)(x^2-4x+16)-x(x-5)(x+5)=264

Jasonbradley

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Jun 16, 2024

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Solving the Equation: (x+4)(x^2-4x+16)-x(x-5)(x+5)=264

This article will guide you through solving the algebraic equation: (x+4)(x^2-4x+16)-x(x-5)(x+5)=264. We'll break down the steps involved in simplifying the equation and ultimately finding the solution(s) for x.

Expanding and Simplifying the Equation

1. Expand the first set of parentheses:

   The expression (x+4)(x^2-4x+16) represents the expansion of a sum of cubes. We can use the formula: (a+b)(a^2-ab+b^2) = a^3 + b^3.

   Therefore, (x+4)(x^2-4x+16) = x^3 + 4^3 = x^3 + 64

2. Expand the second set of parentheses:

   The expression x(x-5)(x+5) involves the difference of squares formula: (a-b)(a+b) = a^2 - b^2.

   Therefore, x(x-5)(x+5) = x(x^2 - 25) = x^3 - 25x

3. Substitute the expanded expressions back into the original equation:

   The equation now becomes: x^3 + 64 - (x^3 - 25x) = 264

4. Simplify by distributing the negative sign:

   x^3 + 64 - x^3 + 25x = 264

5. Combine like terms:

   25x + 64 = 264

Solving for x

1. Isolate the x term:

   25x = 200

2. Divide both sides by 25:

   x = 8

Solution

The solution to the equation (x+4)(x^2-4x+16)-x(x-5)(x+5)=264 is x = 8.