Tewtor.ai

Dividing by 2 is not a separate “step” that changes the equation—it’s simply factoring out the greatest common factor (GCF) from every term so the equation stays equivalent but becomes easier to work with.

Why we look for a GCF first

1. Keeps the equation balanced
   An equation is a statement that two expressions are equal. Whatever you do to one side, you must do to the entire left‑hand side (or to both sides). Dividing the whole left‑hand side by 2 (or multiplying both sides by ½) preserves the equality:

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

   which simplifies to

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

2. Simplifies the numbers
   The coefficients 2, –8, and 6 are all multiples of 2. After dividing, the quadratic becomes

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

   whose leading coefficient is 1. A leading coefficient of 1 makes factoring (or using the quadratic formula) much simpler because you no longer have to keep track of that extra factor.

3. Makes factoring straightforward
   With the simplified form $x^{2}-4x+3$, we only need two numbers that multiply to +3 and add to –4. Those numbers are –1 and –3, giving the factorization

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

   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 quadratic, which adds an extra step and increases the chance of error.

What would happen if we didn’t divide the whole equation?

• Dividing only part of it (e.g., just the constant term) would give an inequivalent equation, leading to a wrong solution.
• Skipping the GCF means you’d have to factor a quadratic with a leading coefficient of 2, which is still possible but more cumbersome: $$2x^{2}-8x+6 = 2(x^{2}-4x+3) = 2(x-1)(x-3)=0,$$ and you’d still end up with the same roots, but the extra factor 2 can distract you.

Bottom line

Dividing by the GCF (here, 2) is a legal algebraic manipulation that:

• preserves the equality,
• reduces the coefficients to their simplest form,
• and makes the subsequent factoring step much clearer.

Does that clarify why we start by dividing the entire equation by 2? Let me know if you’d like to see another example or a visual walkthrough.