(x2-5x+25)(x+5)-x(x-5)(x+5)=75

2 min read Jun 17, 2024
(x2-5x+25)(x+5)-x(x-5)(x+5)=75

Solving the Equation: (x²-5x+25)(x+5) - x(x-5)(x+5) = 75

This problem involves simplifying and solving a polynomial equation. Let's break it down step-by-step:

1. Expanding the Expressions

First, we need to expand the products in the equation:

  • (x²-5x+25)(x+5): Using the distributive property (or FOIL method), we get:
    • x³ + 5x² - 5x² - 25x + 25x + 125 = x³ + 125
  • x(x-5)(x+5): We can use the difference of squares pattern (a² - b² = (a+b)(a-b)) to simplify:
    • x(x² - 25) = x³ - 25x

Now, our equation looks like this: x³ + 125 - (x³ - 25x) = 75

2. Simplifying the Equation

Combining like terms, we get:

x³ + 125 - x³ + 25x = 75

25x + 125 = 75

3. Solving for x

To isolate 'x', we perform the following steps:

  • Subtract 125 from both sides: 25x = -50
  • Divide both sides by 25: x = -2

Therefore, the solution to the equation (x²-5x+25)(x+5) - x(x-5)(x+5) = 75 is x = -2.

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