Tewtor.ai

Why we divide by 2 (the greatest common factor) first

1. Keep the equation equivalent
   Dividing every term of the equation by the same non‑zero number does not change the set of solutions.
   If

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

   then multiplying both sides by ½ (or, equivalently, dividing every term by 2) gives an equation that is exactly the same in terms of the values of $x$ that satisfy it:

   $$\frac{2x^{2}}{2}-\frac{8x}{2}+\frac{6}{2}=0 \;\Longrightarrow\; x^{2}-4x+3=0 .$$

2. Simplify the numbers
   The coefficients become smaller, which makes the next step—factoring—much easier.
   Factoring $x^{2}-4x+3$ only requires finding two numbers that multiply to +3 and add to –4 (‑1 and ‑3).
   If we tried to factor the original $2x^{2}-8x+6$ directly, we would have to consider the extra factor 2 and the larger numbers, which is more cumbersome.

3. Avoid mistakes
   A common error is to divide only part of the equation (for example, just the constant term). That would give an inequivalent equation and lead to wrong solutions. The rule is: if you divide, you must divide every term on both sides by the same number.

4. Preserve the zero‑product property
   After removing the GCF, the quadratic is in a standard form $ax^{2}+bx+c=0$ with $a=1$. This makes it straightforward to apply the zero‑product property once the expression is factored:

   $$(x-1)(x-3)=0 \;\Longrightarrow\; x=1 \text{ or } x=3 .$$

In short:
Dividing by 2 first is a legitimate algebraic step that simplifies the equation while keeping all solutions unchanged. It prevents unnecessary arithmetic and reduces the chance of algebraic slip‑ups.

If you’d like to see another example of factoring after removing a GCF, just let me know!