Solve the equation x² − 5x + 6 = 0

Answer: \(\boxed{x=2 \text{ or } x=3}\)

Factor the quadratic:

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

because \(-2+(-3)=-5\) and \((-2)(-3)=6\).

Now use the zero-product property:

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

So either

\[ x-2=0 \quad\Rightarrow\quad x=2, \]

or

\[ x-3=0 \quad\Rightarrow\quad x=3. \]

Check:

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

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

Both solutions are correct. For similar equations, find two numbers whose sum is the coefficient of \(x\) and whose product is the constant term.