(2x-1)^3+4(x-1)(x+1-2x^2)=7

Jasonbradley

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Jun 16, 2024

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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:

1. Add 5 to both sides: 6x = 12
2. 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.