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Why we factor out the 2 first

The original equation is

All three terms share a common factor of 2. Factoring out that common factor (the greatest common factor, GCF) simplifies the numbers we have to work with, but it does not change the set of solutions because we are multiplying the entire left‑hand side by the same non‑zero number.

• If we divide only one term (e.g., just the first term) by 2, the equation would no longer be equivalent to the original; the balance between the left and right sides would be broken, leading to an incorrect solution.

• If we divide every term (or factor out the 2), we are applying the same operation to the whole left‑hand side. Since 2 ≠ 0, the equation

  $$2\bigl(x^{2}-4x+3\bigr)=0$$

  is true exactly when

  $$x^{2}-4x+3=0$$

  holds. Thus the solutions are unchanged, but the coefficients are smaller and easier to factor.

So the division (or factoring) is a strategic simplification, not a required step for every quadratic. It’s especially helpful when the leading coefficient is larger than 1, as it reduces the numbers you need to consider when looking for factor pairs.

If you’d like a visual, step‑by‑step video that walks through this reasoning, the suggestion above will show exactly how and why we factor out the GCF before solving a quadratic by factoring. Let me know if anything is still unclear!