Solve the equation x² − 5x + 6 = 0

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

Solution

Factor the quadratic:

\[ x^2-5x+6=0 \]

Find two numbers whose:

• product is \(6\);
• sum is \(-5\).

These numbers are \(-2\) and \(-3\). Therefore,

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

So,

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

A product equals zero if at least one factor equals 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 solutions are correct.