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Dividing the whole equation by 2 (the greatest common factor) is a legal algebraic step that makes the quadratic easier to work with, and it does not change the solutions. Here’s why:

1. Equality is preserved when you multiply or divide both sides by the same non‑zero number

If

and we multiply every term on the left‑hand side by the same factor $k\neq0$, the equation stays true:

Because $0/k = 0$, the set of $x$ that satisfy the original equation also satisfy the new one, and vice‑versa.

2. Why choose the greatest common factor (GCF)?

All coefficients (2, –8, 6) share a factor of 2. Dividing by the GCF:

Now the quadratic has smaller, simpler numbers. Factoring $x^{2}-4x+3$ is straightforward because we only need two integers whose product is 3 and whose sum is –4 (‑1 and ‑3). If we tried to factor the original form $2x^{2}-8x+6$ directly, we would have to work with larger numbers and an extra factor of 2, which adds unnecessary steps.

3. What would happen if we divided only one term?

Suppose we divided just the first term by 2:

Now the equation is not equivalent to the original; the set of solutions changes because we altered the balance of the equality. The correct procedure is to divide every term (including the constant term and the right‑hand side) by the same non‑zero number.

4. Summary

• Dividing by the GCF simplifies the coefficients.
• It preserves all solutions because we treat both sides of the equation identically.
• Always apply the division (or multiplication) to every term in the equation.

If you’d like a short visual explanation of why dividing by a common factor works, here’s a helpful video: