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

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

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

This article will guide you through the steps of solving the given equation:

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

1. Factor the denominators:

  • The denominator on the left-hand side can be factored as: (x^2-3x+2) = (x-1)(x-2)
  • The denominator on the right-hand side can be factored as: (x^2-4) = (x+2)(x-2)

2. Rewrite the equation with factored denominators:

This gives us: (1/(x-1)(x-2)) = (1/(x+2)) + (5/(x+2)(x-2))

3. Find a common denominator:

  • The common denominator for all terms is (x-1)(x+2)(x-2).
  • Multiply each term by the appropriate factor to get this common denominator:
    • (1/(x-1)(x-2)) * ((x+2)/(x+2)) = (x+2)/(x-1)(x+2)(x-2)
    • (1/(x+2)) * ((x-1)(x-2)/(x-1)(x-2)) = (x-1)(x-2)/(x-1)(x+2)(x-2)
    • (5/(x+2)(x-2)) * ((x-1)/(x-1)) = 5(x-1)/(x-1)(x+2)(x-2)

4. Combine the terms:

Now the equation becomes: (x+2)/(x-1)(x+2)(x-2) = (x-1)(x-2)/(x-1)(x+2)(x-2) + 5(x-1)/(x-1)(x+2)(x-2)

5. Simplify the equation:

Since all terms have the same denominator, we can combine the numerators: (x+2) = (x-1)(x-2) + 5(x-1)

6. Expand and solve for x:

  • Expand the right-hand side: (x+2) = x^2 - 3x + 2 + 5x - 5
  • Combine like terms: (x+2) = x^2 + 2x - 3
  • Move all terms to one side: 0 = x^2 + x - 5

7. Solve the quadratic equation:

This quadratic equation cannot be easily factored, so we can use the quadratic formula to solve for x:

  • x = (-b ± √(b^2 - 4ac)) / 2a

  • Where a = 1, b = 1, and c = -5

  • x = (-1 ± √(1^2 - 4 * 1 * -5)) / 2 * 1

  • x = (-1 ± √21) / 2

Therefore, the solutions to the equation are:

  • x = (-1 + √21) / 2
  • x = (-1 - √21) / 2

Remember to check for any restrictions on the values of x due to the denominators in the original equation. In this case, x cannot be equal to 1, -2, or 2.

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