(x-4)(x^2+4x+16)+x(x+5)(5-x)=12

3 min read Jun 17, 2024

Solving the Equation (x-4)(x^2+4x+16)+x(x+5)(5-x)=12

This article will guide you through the process of solving the equation (x-4)(x^2+4x+16)+x(x+5)(5-x)=12. We will use algebraic manipulation and simplification techniques to find the solutions for x.

Understanding the Equation

The equation involves a combination of multiplication and addition. The first part, (x-4)(x^2+4x+16), is the product of a linear factor and a quadratic expression. The second part, x(x+5)(5-x), is the product of three linear factors. Our goal is to simplify the equation and solve for the unknown variable, x.

Solving the Equation

Expand the products:

Begin by expanding the products using the distributive property.
- (x-4)(x^2+4x+16) = x³ + 4x² + 16x - 4x² - 16x - 64 = x³ - 64
- x(x+5)(5-x) = x(5x - x² + 25 - 5x) = -x³ + 25x
Combine like terms:

Substitute the expanded expressions back into the original equation:

x³ - 64 - x³ + 25x = 12

Simplify by combining like terms:

25x - 64 = 12
Isolate the variable:

Add 64 to both sides of the equation:

25x = 76
Solve for x:

Divide both sides by 25:

x = 76/25

Solution

Therefore, the solution to the equation (x-4)(x^2+4x+16)+x(x+5)(5-x)=12 is x = 76/25.

Conclusion

By systematically expanding, simplifying, and isolating the variable, we successfully solved the equation. Remember to always check your solution by substituting it back into the original equation to ensure its validity.