x^2-5x+6=0

To solve the quadratic equation \(x^2 - 5x + 6 = 0\), we can use factoring, the quadratic formula, or completing the square. In this case, the equation can be factored easily.

We want to find two numbers that multiply to \(6\) (the constant term) and add to \(-5\) (the coefficient of the linear term).

The numbers \(-2\) and \(-3\) satisfy these conditions, since \((-2) \cdot (-3) = 6\) and \((-2) + (-3) = -5\).

Thus, we can factor the equation as:

\[ (x - 2)(x - 3) = 0 \]

Setting each factor equal to zero gives us the solutions:

1. \(x - 2 = 0 \quad \Rightarrow \quad x = 2\)
2. \(x - 3 = 0 \quad \Rightarrow \quad x = 3\)

Therefore, the solutions to the equation \(x^2 - 5x + 6 = 0\) are:

\[ x = 2 \quad \text{and} \quad x = 3 \]