%5E3%2B4(x-1)(x%2B1-2x%5E2)%3D7.jpeg "(2x-1)^3+4(x-1)(x+1-2x^2)=7")
Solving the Equation: (2x-1)^3 + 4(x-1)(x+1-2x^2) = 7
This article will guide you through the process of solving the equation (2x-1)^3 + 4(x-1)(x+1-2x^2) = 7.
Expanding and Simplifying
First, we need to expand and simplify the equation. We'll use the following identities:
- (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
- (a - b)(a + b) = a^2 - b^2
Let's start by expanding the first term:
(2x-1)^3 = (2x)^3 + 3(2x)^2(-1) + 3(2x)(-1)^2 + (-1)^3 = 8x^3 - 12x^2 + 6x - 1
Next, we'll expand the second term:
4(x-1)(x+1-2x^2) = 4(x^2 - 1 - 2x^3 + 2x^2) = -8x^3 + 12x^2 - 4
Now, we can substitute these expansions back into the original equation:
(8x^3 - 12x^2 + 6x - 1) + (-8x^3 + 12x^2 - 4) = 7
Simplifying the equation, we get:
6x - 5 = 7
Solving for x
Now, we have a simple linear equation. Let's solve for x:
- Add 5 to both sides: 6x = 12
- Divide both sides by 6: x = 2
Therefore, the solution to the equation (2x-1)^3 + 4(x-1)(x+1-2x^2) = 7 is x = 2.