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Why we divide (or factor) by 2 first

1. Preserve the equality
   An equation stays true only if every term on both sides is treated the same way.
   Dividing the whole left‑hand side by 2 (or factoring a 2 out) is equivalent to multiplying the entire equation by ½. That operation does not change the set of solutions because we are applying the same factor to every term.

2. Simplify the numbers
   The original quadratic has coefficients 2, –8, +6.
   After dividing by the GCF 2 we get the simpler quadratic

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

   Smaller numbers are easier to work with when you look for factors or apply the quadratic formula.

3. Avoid mistakes
   A common error is to divide only one term (e.g., turning $2x^{2} - 8x + 6$ into $x^{2} - 8x + 6$). That changes the balance of the equation and leads to wrong solutions. The correct step is to divide all three terms (or factor the 2 out) so the equation remains equivalent.

4. The factor 2 itself can’t be zero
   After factoring, the equation looks like

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

   Since the constant factor 2 is never zero, the only way the product can be zero is if the quadratic inside the parentheses is zero. Hence we solve

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

   This reduction is what lets us factor it easily as $(x-1)(x-3)=0$.

Quick check

If we didn’t divide every term, the equation would become something like

which is a different problem with different roots. That’s why the GCF step must apply to the whole expression.

Want a visual explanation?
A short video walks through the idea of factoring out the greatest common factor before solving a quadratic.

[The video will appear here]