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Answer

We divide the entire equation by 2 (the greatest common factor) to simplify it while keeping the equality true.

Why we factor out the 2

1. Maintain balance – An equation is a statement that two expressions are equal.
   If you multiply or divide both sides by the same non‑zero number, the equality remains valid.

2. Simplify the numbers – Every term in

   $$2x^{2} - 8x + 6 = 0$$

   contains a factor of 2. Removing that common factor makes the coefficients smaller and the subsequent factoring step easier.

3. How it works
   Divide each term on the left side (and the 0 on the right, which stays 0) by 2:

   $$\frac{2x^{2}}{2} \;-\; \frac{8x}{2} \;+\; \frac{6}{2} = \frac{0}{2}$$

   which simplifies to

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

   The factor 2 outside the parentheses can be written as

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

   and since 2 is never zero, the equation reduces to the simpler quadratic inside the parentheses.

4. What would happen if you divided only one term?
   Changing just one term (e.g., turning $2x^{2}$ into $x^{2}$ but leaving $-8x + 6$ unchanged) would alter the relationship between the left‑hand side and the right‑hand side, producing a different equation and generally leading to an incorrect solution.

Try it yourself

You can explore how the quadratic changes as you factor out different numbers with the interactive “quadratic‑explorer” tool:

Bottom line

Dividing by the GCF 2 every term simplifies the problem without breaking the equality, making the factoring step straightforward and ensuring the solutions you find are the true solutions of the original equation.