(x%2Bx%5E2)%2B2(x-5)(x%2B1)-x%5E3%3D12.jpeg "(x-3)(x+x^2)+2(x-5)(x+1)-x^3=12")
Solving the Equation (x-3)(x+x^2)+2(x-5)(x+1)-x^3=12
This article will guide you through the steps to solve the equation (x-3)(x+x^2)+2(x-5)(x+1)-x^3=12. Let's break it down:
1. Expand the Products
First, we need to expand the products in the equation. Remember to use the distributive property (or FOIL method):
-
(x-3)(x+x^2):
- x * (x+x^2) = x^2 + x^3
- -3 * (x+x^2) = -3x - 3x^2
- So, (x-3)(x+x^2) = x^2 + x^3 - 3x - 3x^2
-
2(x-5)(x+1):
- 2 * (x-5)(x+1) = 2 * (x^2 - 4x - 5) = 2x^2 - 8x - 10
Now, let's substitute these expanded terms back into the original equation:
(x^2 + x^3 - 3x - 3x^2) + (2x^2 - 8x - 10) - x^3 = 12
2. Combine Like Terms
Next, combine all the terms with the same powers of 'x':
- x^3: x^3 - x^3 = 0
- x^2: x^2 - 3x^2 + 2x^2 = 0
- x: -3x - 8x = -11x
- Constant: -10 = -10
This simplifies our equation to: -11x - 10 = 12
3. Isolate the 'x' term
To solve for 'x', we need to isolate the 'x' term. Add 10 to both sides of the equation:
-11x = 22
4. Solve for 'x'
Finally, divide both sides of the equation by -11 to find the value of 'x':
x = -2
Conclusion
Therefore, the solution to the equation (x-3)(x+x^2)+2(x-5)(x+1)-x^3=12 is x = -2.