Solve the equation x² − 5x + 6 = 0

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

Key steps:

1. Find two numbers whose product is \(6\) and whose sum is \(-5\).
   These numbers are \(-2\) and \(-3\).

2. Factor the quadratic: \[ x^{2}-5x+6=(x-2)(x-3) \]

3. Use the zero-product property: \[ (x-2)(x-3)=0 \] So either \[ x-2=0 \quad \text{or} \quad x-3=0 \] Hence \[ x=2 \quad \text{or} \quad x=3. \]

Check:

• For \(x=2\): \(2^{2}-5\cdot2+6=4-10+6=0\).
• For \(x=3\): \(3^{2}-5\cdot3+6=9-15+6=0\).

Both work. The principle is: if a product equals zero, at least one factor must equal zero.