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When we solve an equation we must keep the two sides exactly equal.
Dividing every term on both sides by the same non‑zero number preserves that equality, just as multiplying both sides by the same number does.

Why we look for a greatest common factor (GCF)

The quadratic

has the numbers 2, 8 and 6 as coefficients. All three share a common factor of 2.
If we factor that 2 out of the whole left‑hand side, we get

Now we can divide the entire equation by 2 (the same factor that we factored out) to obtain an equivalent, simpler equation:

What would happen if we divided only part of the equation?

Suppose we tried to divide just the first term:

The left‑hand side is no longer the same expression we started with, so the new equation is not equivalent to the original one. Its solutions would be different (in fact, $x^{2}-8x+6=0$ has roots $x=1$ and $x=6$, which are not the correct roots of the original problem).

Benefits of dividing by the GCF first

1. Simplifies coefficients – smaller numbers are easier to work with when factoring or applying the quadratic formula.
2. Reduces arithmetic errors – fewer large numbers mean fewer chances to slip up.
3. Keeps the equation equivalent – because we divide every term (including the constant term) by the same non‑zero factor, the set of solutions stays exactly the same.

So we divide by 2 first to obtain the simpler, equivalent quadratic $x^{2}-4x+3=0$, which we can then factor easily:

Feel free to ask if any part of this reasoning is still unclear!