Tewtor.ai

Why we factor out the 2 (divide the whole equation by the GCF)

1. Preserves the equality
   When we “divide by 2” we are really doing this:

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

   Because we divide every term on both sides by the same non‑zero number, the two sides stay equal. The set of solutions does not change.

2. Simplifies the numbers
   After removing the common factor 2, the quadratic becomes

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

   The coefficients are now smaller, making it easier to spot the pair of numbers that multiply to +3 and add to ‑4 (‑1 and ‑3). If we tried to factor the original form directly, we would have to work with the larger numbers 2, ‑8, and 6, which is more cumbersome.

3. Helps avoid mistakes
   A common error is to divide only one term (e.g., turning $2x^{2}$ into $x^{2}$ but leaving $‑8x$ unchanged). That creates a different equation with different solutions. By factoring the GCF, we are forced to apply the division to all terms, guaranteeing the transformed equation is equivalent to the original.

4. Doesn’t affect the roots
   If $x$ satisfies the original equation, it also satisfies the reduced one, and vice‑versa. For example, $x=1$ makes

   $$2(1)^{2}-8(1)+6 = 2-8+6 = 0$$

   and it also makes

   $$(1)^{2}-4(1)+3 = 1-4+3 = 0$$

   So the roots are unchanged.

Bottom line:
We factor out the greatest common factor (the 2) because it makes the quadratic easier to factor while keeping the equation mathematically equivalent. It’s a safe, systematic first step that prevents the “divide only the first term” mistake you’ve encountered before.

Would you like me to walk through another example, or clarify any part of this reasoning?