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

3 min read Jun 16, 2024

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

This article explores the solution to the equation (2x-1)(x^2-x+1) = 2x^3-3x^2+2. We will analyze the equation, expand it, and determine its solutions.

Expanding the Left Side of the Equation

First, we need to expand the left side of the equation using the distributive property:

(2x-1)(x^2-x+1) = 2x(x^2-x+1) - 1(x^2-x+1)

Simplifying further:

= 2x^3 - 2x^2 + 2x - x^2 + x - 1

= 2x^3 - 3x^2 + 3x - 1

Comparing Expanded Forms

Now, we have the expanded form of the left side: 2x^3 - 3x^2 + 3x - 1. Let's compare it to the right side of the original equation: 2x^3 - 3x^2 + 2.

Notice that the first two terms (2x^3 and -3x^2) are identical on both sides. However, the constant terms and the 'x' terms differ.

Analyzing the Discrepancy

The discrepancy arises from the '3x' term on the expanded left side and the absence of an 'x' term on the right side. This indicates that the equation is not an identity. In other words, the equation is only true for certain values of 'x'.

Finding the Solutions

To find the solutions of the equation, we need to set the expanded forms equal to each other and solve for 'x':

2x^3 - 3x^2 + 3x - 1 = 2x^3 - 3x^2 + 2

Simplifying by canceling out the common terms:

3x - 1 = 2

Solving for 'x':

3x = 3

x = 1

Conclusion

Therefore, the only solution to the equation (2x-1)(x^2-x+1) = 2x^3-3x^2+2 is x = 1. This means that the equation holds true only when the value of 'x' is 1.