Final answer:
To solve the given algebraic equation, simplify and eliminate fractions, then apply the quadratic formula or use a graphing calculator for higher-degree polynomials. Approximations may be used if a term is negligible compared to others.
Step-by-step explanation:
The student's question is about solving an algebraic equation. To solve the equation (1/6 ((1/3)*x)) = ((1/3) ((1/3)*x))/(1-x²), it's essential to start by simplifying both sides and eliminating fractions, if possible. Let's multiply both sides by the least common denominator to clear out the fractions.
After simplification, you might end up with a quadratic equation or sometimes a higher degree polynomial, depending on the specific terms in your equation. If it's a quadratic equation, as indicated by a form similar to ax²+bx+c=0, the quadratic formula can be applied to find the values of x. However, if the equation has a higher degree and doesn't simplify to a quadratic form, a graphing calculator might be necessary to find the zeros or approximate solutions.
In specific scenarios, if a term in the equation is very small relative to others (an assumption you must confirm is valid for your case), you might simplify the equation by neglecting the small term. This approximation can make it much simpler to solve. Finally, after solving the equation, always remember to check back to ensure that the solutions satisfy the original equation without making the denominator zero in case of fractions.