Solve: x² − 5x + 6 = 0

Solve quadratic equations by factoring when possible. Step 1: factor — find numbers that multiply to 6 and add to −5: that's −2 and −3 (since (−2)(−3) = 6 and −2 + −3 = −5). So x² − 5x + 6 = (x − 2)(x − 3). Step 2: set the factored form equal to zero: (x − 2)(x − 3) = 0. Step 3: zero-product property — if a product is zero, at least one factor is zero. So x − 2 = 0 (giving x = 2) or x − 3 = 0 (giving x = 3). Solutions: x = 2 or x = 3. Verify: 2² − 5(2) + 6 = 4 − 10 + 6 = 0 ✓; 3² − 5(3) + 6 = 9 − 15 + 6 = 0 ✓. A quadratic equation can have 0, 1, or 2 real solutions. If factoring doesn't work, use the quadratic formula: x = (−b ± √(b² − 4ac)) / (2a) for ax² + bx + c = 0. The discriminant b² − 4ac: positive → 2 solutions; zero → 1 solution; negative → no real solutions.