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

Jasonbradley

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Jun 17, 2024

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

This equation presents a rational function set equal to zero. To solve for x, we can utilize the following steps:

Understanding the Equation

• Numerator: The numerator is factored into (x+2) and (x^(2)-2x+1). The second factor is a perfect square trinomial: (x-1)^(2)
• Denominator: The denominator is a quadratic expression. We can factor it as well.

Solving for x

1. Set the numerator equal to zero: Since a fraction equals zero only when the numerator is zero, we have: (x+2)(x-1)^(2) = 0
2. Solve for x: This equation has two solutions:
   • x + 2 = 0 => x = -2
   • (x-1)^(2) = 0 => x = 1

Verifying the Solutions

It's crucial to check if these solutions make the denominator equal to zero. If they do, they are extraneous solutions and must be discarded.

1. For x = -2: 4 + 3(-2) - (-2)^(2) = 4 - 6 - 4 = -6 ≠ 0. Therefore, x = -2 is a valid solution.
2. For x = 1: 4 + 3(1) - (1)^(2) = 4 + 3 - 1 = 6 ≠ 0. Therefore, x = 1 is a valid solution.

Conclusion

The solutions to the equation ((x+2)(x^(2)-2x+1))/(4+3x-x^(2)) = 0 are x = -2 and x = 1. These solutions have been verified and are not extraneous.