Final answer:
The expression x²-5x+6/x²-9 simplifies to (x-2)/(x+3) after factoring the numerator as (x-2)(x-3) and the denominator as (x-3)(x+3), and then canceling the common factor (x-3).
Step-by-step explanation:
The expression x²-5x+6/x²-9 can be simplified by factoring both the numerator and the denominator. The numerator factors into (x-2)(x-3) and the denominator factors into (x-3)(x+3), as both are difference of squares or simple quadratic trinomials. After factoring, we can simplify the expression by canceling out the common factors. The simplified expression is therefore (x-2)/(x+3).
To factor the numerator, we look for two numbers that multiply to 6 and add up to -5, which are -2 and -3. For the denominator, we recognize that x²-9 is a difference of squares, which can be factored into (x+3)(x-3). We then divide both the numerator and the denominator by the common factor (x-3), which gives us the simplified result.