Solve the equation x² − 5x + 6 = 0

Answer: \(x=2\) or \(x=3\).

Solution

Factor the quadratic:

\[ x^2-5x+6=(x-2)(x-3) \]

We chose \(-2\) and \(-3\) because:

\[ (-2)+(-3)=-5,\qquad (-2)(-3)=6 \]

Therefore,

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

A product equals zero if at least one factor is zero:

\[ x-2=0 \quad \text{or} \quad x-3=0 \]

Hence,

\[ \boxed{x=2 \text{ or } x=3} \]

Check:

\[ 2^2-5\cdot2+6=4-10+6=0 \]

\[ 3^2-5\cdot3+6=9-15+6=0 \]

Both roots are correct.