- Identify restrictions: $x \neq 5$ and $x \neq -5$ (denominators cannot be zero).
- Multiply both sides by the common denominator $(x^2 - 5)(x + 5)(x - 5)$ to clear fractions.
- Simplify the resulting equation: $(x - 3)(x + 5)(x - 5) + (x^2 - 5)(x - 5) = (x^2 - 5)(x + 5)$.
- Expand all terms: $(x - 3)(x^2 - 25) + (x^3 - 5x^2 - 5x + 25) = x^3 + 5x^2 - 5x - 25$.
- Combine like terms on the left: $x^3 - 3x^2 - 25x + 75 + x^3 - 5x^2 - 5x + 25 = 2x^3 - 8x^2 - 30x + 100$.
- Set equal to right side: $2x^3 - 8x^2 - 30x + 100 = x^3 + 5x^2 - 5x - 25$.
- Move all terms to one side: $x^3 - 13x^2 - 25x + 125 = 0$.
- Factor the cubic: $(x - 5)(x^2 - 8x - 25) = 0$.
- Solve each factor: $x = 5$ or $x = 4 \pm \sqrt{41}$.
- Reject $x = 5$ as it violates the restriction.
- Final solution: $x = 4 + \sqrt{41}$ or $x = 4 - \sqrt{41}$.
Parent Tip: Review the logic above to help your child master the concept of solving rational equations.
