(x%2B5)-x(x-5)(x%2B5)%3D75.jpeg "(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.