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Dividing the entire equation by 2 is a legal algebraic operation because you are multiplying both sides of the equation by the same non‑zero number (½).

What “divide the whole equation” means

The original equation is

If you multiply every term on the left‑hand side by ½, you must also multiply the right‑hand side (the 0) by ½:

Carrying out the division gives

Because 0 ÷ 2 = 0, the right‑hand side stays 0, and the set of solutions does not change.

Why we do it

1. Simplify the numbers – Smaller coefficients make it easier to spot factor pairs (here, –1 and –3) or to apply the quadratic formula without large fractions.
2. Avoid arithmetic errors – Working with $x^{2}-4x+3$ reduces the chance of mis‑calculating products or sums during factoring.
3. Preserve equivalence – Multiplying or dividing both sides by the same non‑zero constant yields an equivalent equation; it has exactly the same solution set.

Common mistake to avoid

Sometimes students divide only part of the left‑hand side, e.g., turning $2x^{2}-8x+6$ into $x^{2}-8x+3$. That changes the equation’s balance because the right‑hand side (0) is not treated the same way, and the solutions would be different. Always apply the operation to every term on both sides.

Bottom line: Dividing by 2 simplifies the quadratic while keeping the solutions unchanged, as long as the division is applied to the entire equation (both sides).

If you’d like another visual explanation of this step, let me know and I can suggest a different video.