Final answer:
To solve the given rational equation, factor the denominators, find a common denominator, combine the fractions, simplify to get a quadratic equation, and then use the quadratic formula to solve for x.
Step-by-step explanation:
To solve the rational equation 5/( x² -x-6) = 2- (x-3)/(x-2), we first need to factor the denominators and identify the common denominator to combine the fractions. The denominator x² - x - 6 can be factored as (x - 3)(x + 2). Hence, our common denominator is (x - 2)(x - 3)(x + 2). We then multiply both sides of the equation by this common denominator and simplify the resulting equation.
Upon simplification, we obtain a quadratic equation which can be solved using the quadratic formula. The general form of a quadratic equation is ax² + bx + c = 0, and the quadratic formula is x = (-b ± √(b² - 4ac))/(2a). By substituting the coefficients from our simplified equation into the quadratic formula, we can find the solution(s) for x.
Remember to check for extraneous solutions by substituting them back into the original equation, as some solutions may not be valid because they make the denominators zero.