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(3x^2-x+1)(x-1)+x^2(4-3x)=5/2

2 min read Jun 16, 2024
(3x^2-x+1)(x-1)+x^2(4-3x)=5/2

Solving the Equation: (3x^2-x+1)(x-1)+x^2(4-3x)=5/2

This article will guide you through the steps of solving the given equation: (3x^2-x+1)(x-1)+x^2(4-3x)=5/2.

1. Expanding the Equation

First, we need to expand the equation by multiplying out the terms:

  • (3x^2-x+1)(x-1): This involves using the distributive property (or FOIL method)
    • 3x^2(x-1) = 3x^3 - 3x^2
    • -x(x-1) = -x^2 + x
    • 1(x-1) = x - 1
  • x^2(4-3x) = 4x^2 - 3x^3

Now, the equation becomes: 3x^3 - 3x^2 - x^2 + x + x - 1 + 4x^2 - 3x^3 = 5/2

2. Simplifying the Equation

Combine like terms on the left side of the equation:

(-3x^3 + 3x^3) + (-3x^2 - x^2 + 4x^2) + (x + x) - 1 = 5/2

This simplifies to: 0 + 0 + 2x - 1 = 5/2

3. Isolating the Variable

To solve for x, we need to isolate it on one side of the equation.

  • 2x - 1 = 5/2
  • 2x = 5/2 + 1
  • 2x = 7/2

4. Solving for x

Finally, divide both sides of the equation by 2 to find the value of x:

  • x = (7/2) / 2
  • x = 7/4

Therefore, the solution to the equation (3x^2-x+1)(x-1)+x^2(4-3x)=5/2 is x = 7/4.

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