Correct Answer:
A. 2,3
The correct answer is A: 2, 3. To find the roots of the quadratic equation x² - 5x + 6 = 0, we can use factorization, the quadratic formula, or by testing the given options. Factorization is often the quickest method for simpler quadratics.
- We need to find two numbers that multiply to +6 (the constant term) and add up to -5 (the coefficient of the x term). These numbers are -2 and -3.
- So, we can factor the equation as (x - 2)(x - 3) = 0.
- For the product of two factors to be zero, at least one of the factors must be zero.
- Set the first factor to zero: x - 2 = 0 → x = 2.
- Set the second factor to zero: x - 3 = 0 → x = 3.
- Therefore, the roots of the equation are 2 and 3.
Let's examine why the other options are incorrect:
- Option B: 1, 6 is incorrect. If we substitute these values into the equation:
- For x = 1: (1)² - 5(1) + 6 = 1 - 5 + 6 = 2 ≠ 0.
- For x = 6: (6)² - 5(6) + 6 = 36 - 30 + 6 = 12 ≠ 0.
- Option C: -2, -3 is incorrect. If we substitute these values:
- For x = -2: (-2)² - 5(-2) + 6 = 4 + 10 + 6 = 20 ≠ 0.
- For x = -3: (-3)² - 5(-3) + 6 = 9 + 15 + 6 = 30 ≠ 0. (Also, (x+2)(x+3) = x² + 5x + 6, not x² - 5x + 6).
- Option D: 0, 6 is incorrect.
- For x = 0: (0)² - 5(0) + 6 = 6 ≠ 0.
- For x = 6, as shown above, it results in 12 ≠ 0.