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(x+2)(x^2-2x+1)/4+3x-x^2 =0

2 min read Jun 16, 2024
(x+2)(x^2-2x+1)/4+3x-x^2 =0

Solving the Equation: (x+2)(x^2-2x+1)/4 + 3x - x^2 = 0

This equation involves a fraction, a quadratic term, and linear terms. To solve it, we'll follow these steps:

  1. Simplify the equation:

    • Expand the numerator of the fraction: (x+2)(x^2-2x+1) = x^3 - 2x^2 + x + 2x^2 - 4x + 2 = x^3 - 3x + 2
    • Rewrite the equation: (x^3 - 3x + 2)/4 + 3x - x^2 = 0

  2. Multiply both sides by 4:

    • This gets rid of the fraction: x^3 - 3x + 2 + 12x - 4x^2 = 0

  3. Combine like terms:

    • Rearrange the terms in descending order of their exponents: x^3 - 4x^2 + 9x + 2 = 0

  4. Solve for x: This equation is a cubic equation, which can be difficult to solve directly. We can try the following methods:

    • Factoring: Try to factor the cubic expression. However, this cubic equation doesn't have simple integer factors.
    • Rational Root Theorem: This theorem helps find potential rational roots. But, in this case, it won't lead to easy solutions.
    • Numerical Methods: Use numerical methods like the Newton-Raphson method or graphing calculators to find approximate solutions.

Therefore, finding the exact solutions for this equation is challenging. You can use numerical methods or graphing calculators to obtain approximate solutions.

Important Note: Remember to check your solutions by plugging them back into the original equation to ensure they are valid.

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