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

If

then for any non‑zero constant $k$,

holds true, because you are performing the same operation on the left‑hand side (LHS) and the right‑hand side (RHS). The set of solutions does not change.

Why it helps here

1. Smaller numbers – After dividing by 2 the coefficients become 1, –4, 3 instead of 2, –8, 6. Smaller numbers are easier to work with when you:

   • look for factors,
   • compute the discriminant,
   • apply the quadratic formula.

2. Factoring becomes obvious – The quadratic $x^{2}-4x+3$ factors cleanly as $(x-3)(x-1)$. With the original coefficients you would have to factor $2x^{2}-8x+6$, which still works but requires an extra step (pulling out a common factor first).

3. No loss of solutions – Because 2 ≠ 0, dividing by 2 cannot introduce or discard any roots. Any $x$ that satisfies the simplified equation also satisfies the original one, and vice‑versa.

You could also solve it without dividing

If you preferred to keep the original form, you could apply the quadratic formula directly with $a=2$, $b=-8$, $c=6$:

which again gives $x=3$ or $x=1$. The result is identical; the division step merely makes the arithmetic a bit cleaner.

Bottom line: Dividing by 2 is a convenient simplification, not a required step. It preserves the equality and often makes the subsequent algebra faster and less error‑prone.

Let me know if you’d like a visual walkthrough of the quadratic‑formula steps or any other clarification!