Tewtor.ai

Dividing by 2 is not required to solve the equation, but it makes the algebra much simpler. Here’s why we chose that step:

1. Identify a common factor
   Every term in the quadratic $2x^{2}-8x+6$ shares the factor 2:

   $$2x^{2}=2\cdot x^{2},\qquad -8x=2\cdot(-4x),\qquad 6=2\cdot 3$$

   When a number (the greatest common factor, GCF) multiplies every term, we can factor it out of the whole left‑hand side.

2. Factor the GCF out of the entire equation

   $$2x^{2}-8x+6 = 2\bigl(x^{2}-4x+3\bigr)$$

   Because the right‑hand side of the original equation is 0, we can divide both sides of the equation by the same non‑zero number (here, 2) without changing the set of solutions:

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

3. Why this helps

   • The coefficients become smaller (1, –4, 3 instead of 2, –8, 6), so factoring or applying the quadratic formula is easier and less error‑prone.
   • It eliminates the extra step of carrying a factor of 2 through the later algebra, which is a common source of mistakes (e.g., forgetting to divide a term when you expand or solve).

4. Important rule:
   The division must be applied to the entire equation, not just to one term. If you only divide part of the expression, the equality no longer holds and you’ll obtain incorrect roots. For example, dividing only the first two terms would give

   $$x^{2}-4x+6=0,$$

   which is a different equation with different solutions.

So we divided by 2 first because it is the GCF of all terms, and doing so simplifies the quadratic while preserving the same solutions. After the division we factored the simpler quadratic $(x-1)(x-3)=0$ and found the roots $x=1$ and $x=3$.

Let me know if you’d like to see the same problem solved without first dividing by the GCF, or if you’d like more practice spotting and using GCFs in other equations!