(x-1)^3-x(x+1)^2=5x(2-x)-11(x+2)

4 min read Jun 17, 2024

Solving the Equation: (x-1)^3 - x(x+1)^2 = 5x(2-x) - 11(x+2)

This article will guide you through the process of solving the equation (x-1)^3 - x(x+1)^2 = 5x(2-x) - 11(x+2). We will break down each step, making it easy to follow.

Expanding the Equation

First, we need to expand the equation to get rid of the parentheses and simplify it.

Expanding the cubes and squares:
- (x-1)^3 = (x-1)(x-1)(x-1) = x^3 - 3x^2 + 3x - 1
- x(x+1)^2 = x(x+1)(x+1) = x(x^2 + 2x + 1) = x^3 + 2x^2 + x
Expanding the remaining products:
- 5x(2-x) = 10x - 5x^2
- -11(x+2) = -11x - 22

Now, our equation becomes: x^3 - 3x^2 + 3x - 1 - (x^3 + 2x^2 + x) = 10x - 5x^2 - 11x - 22

Simplifying the Equation

Next, we combine like terms and simplify:

Distributing the negative sign: x^3 - 3x^2 + 3x - 1 - x^3 - 2x^2 - x = 10x - 5x^2 - 11x - 22
Combining like terms: -5x^2 + 2x - 1 = -x^2 - x - 22
Moving all terms to one side: -5x^2 + 2x - 1 + x^2 + x + 22 = 0
Combining like terms again: -4x^2 + 3x + 21 = 0

Solving the Quadratic Equation

We now have a quadratic equation. There are a couple of ways to solve it:

Factoring: Try to find two numbers that multiply to give -84 (-4 * 21) and add to give 3. The numbers 12 and -7 work:
- -4x^2 + 12x - 7x + 21 = 0
- 4x(x - 3) - 7(x - 3) = 0
- (4x - 7)(x - 3) = 0
- Therefore, x = 3 or x = 7/4
Quadratic Formula: If factoring doesn't work easily, use the quadratic formula:
- x = [-b ± √(b^2 - 4ac)] / 2a
- In our case, a = -4, b = 3, and c = 21.
- Plugging these values into the formula and simplifying, we get the same solutions as before: x = 3 or x = 7/4.

Conclusion

The solutions to the equation (x-1)^3 - x(x+1)^2 = 5x(2-x) - 11(x+2) are x = 3 and x = 7/4.