From factoring quadratics to first isomorphism theorem.

Solve x² − 5x + 6 = 0 two ways

Two valid routes to the same roots — and a reason to prefer one of them under exam pressure.

1. 1
   $$x^2 - 5x + 6 = (x-2)(x-3)$$

   Look for a factorisation first: two numbers multiplying to 6 and adding to −5.

2. 2
   $$x = 2 \quad\text{or}\quad x = 3$$

   A product is zero only when one factor is zero — this is the reasoning, not a shortcut.

3. 3
   $$x = \frac{5 \pm \sqrt{25 - 24}}{2}$$

   The quadratic formula always works, so it is the fallback when no factorisation is visible.

4. 4
   $$x = \frac{5 \pm 1}{2} = 3,\ 2$$

   Same roots. The discriminant of 1 also tells us the roots are real, distinct and rational.

Why this matters — Knowing why factoring works is what lets a student choose the faster method instead of defaulting to one.