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When we “divide by 2 first,” we’re factoring out the greatest common factor (GCF) from every term of the equation.

Why we do it

1. All terms share the same factor
   In

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

   each term ( $2x^{2}$, $-8x$, $+6$ ) is divisible by 2.

2. Dividing the whole equation keeps it balanced
   An equation is a statement that two sides are equal. If you divide only one side or only some terms, the equality is broken. To preserve the equality, you must divide every term on both sides by the same non‑zero number.

3. Simplifies the numbers
   Removing the common factor reduces the coefficients, making the quadratic easier to factor or to apply the quadratic formula.

   After dividing every term by 2:

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

4. Does not change the solutions
   Because we performed an algebraic operation that is reversible (multiply both sides by 2 to get back the original equation), the set of solutions remains exactly the same.

Common mistake

If you divide only part of the equation—say, just the first term—you’d get something like

which is not equivalent to the original equation; the solutions would be different. Always apply the division (or any operation) to all terms on both sides of the equation.

Quick recap of the full solution

1. Factor out the GCF (divide every term by 2): $x^{2} - 4x + 3 = 0$
2. Factor the quadratic: $(x-1)(x-3)=0$
3. Set each factor to zero: $x=1$ or $x=3$

If you’d like a visual explanation of factoring quadratics and the role of the GCF, the video below walks through the process step by step.