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Dividing by 2 isn’t required to solve the quadratic, but it makes the algebra much easier and reduces the chance of arithmetic errors. Here’s why:

1. Preserve the equality
   When we divide an equation by a non‑zero number, we must divide every term on both sides by that same number.

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

   which simplifies to

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

   The new equation is exactly equivalent to the original one; any solution of the simplified equation is also a solution of the original.

2. Simplify the coefficients
   The original quadratic has coefficients 2, –8, and 6. After dividing by the greatest common factor (GCF = 2) the coefficients become 1, –4, and 3. Working with smaller numbers:

   • makes mental arithmetic faster,
   • reduces the risk of sign mistakes,
   • and often reveals factor pairs more readily.

3. Easier factoring
   Factoring $x^{2}-4x+3$ is straightforward because we look for two numbers whose product is +3 and whose sum is –4. Those numbers are –1 and –3, giving

   $$(x-1)(x-3)=0 .$$

   If we tried to factor the original $2x^{2}-8x+6$ directly, we would have to consider a factor of 2 in front of the whole expression or search for pairs that multiply to $2\times6=12$ while summing to –8, which is more cumbersome.

4. No loss of solutions
   Because we divided by a non‑zero constant (2), we did not introduce or discard any roots. The solution set of the simplified equation $\{1,3\}$ is exactly the solution set of the original equation.

In short: dividing by the GCF (2) keeps the equation equivalent while turning a “hard‑to‑factor” quadratic into a simpler one, making the factoring step quick and reliable. If you ever skip the division and try to factor the original coefficients, you’re more likely to miss the correct factor pair or make an arithmetic slip.