What is the factorization of the quadratic \( x^2 - 5x + 6 \)?

To factor the quadratic ( x^2 - 5x + 6 ), we look for two numbers that multiply to the constant term (6) and add to the coefficient of the linear term (-5).

The numbers that meet these criteria are -2 and -3. These two numbers multiply together to give 6 (since (-2 \times -3 = 6)) and add up to -5 (since (-2 + -3 = -5)).

Using these numbers, we can express the quadratic in its factored form. The factorization is ( (x - 2)(x - 3) ). To verify, we can expand this expression:

[

(x - 2)(x - 3) = x^2 - 3x - 2x + 6 = x^2 - 5x + 6,

]

which matches the original quadratic. Therefore, the correct factorization of ( x^2 - 5x + 6 ) is indeed ( (x - 2)(x - 3) ).

Understanding the method of factoring quadratics involves recognizing patterns and coefficients, and applying this methodology can help in solving similar problems efficiently