Tewtor.ai

Dividing the entire equation by the greatest common factor (GCF) is a legal algebraic operation that makes the quadratic easier to factor, and it never changes the set of solutions.

Why we can divide by 2

The original equation is

Every term on the left‑hand side shares the factor 2:

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

If we divide both sides of the equation by the same non‑zero number (here, 2), we are performing the same operation on the entire equation, so the equality remains true. In other words,

Because we multiplied (or divided) both sides by the same factor, any value of $x$ that satisfies the new, simpler equation will also satisfy the original one, and vice‑versa.

Practical benefit

The simplified quadratic $x^{2} - 4x + 3 = 0$ has leading coefficient 1, which makes it much easier to factor:

• We look for two numbers whose product is $+3$ and whose sum is $-4$.
• Those numbers are $-1$ and $-3$, giving $(x-1)(x-3)=0$.

If we tried to factor the original form $2x^{2} - 8x + 6$ directly, we would have to consider a leading coefficient of 2, which adds an extra step (splitting the middle term with a factor of 2). Dividing by the GCF removes that extra complication.

Important caution

Divide the entire equation, not just one term.
If you only divided the first term by 2, you would get an inconsistent equation (e.g., $x^{2} - 8x + 6 = 0$), which leads to wrong solutions. The division must be applied to every term on both sides.

Bottom line:

• Dividing by the GCF simplifies the coefficients.
• It preserves all solutions because we treat both sides equally.
• Always apply the division to the whole equation, not just part of it.

Let me know if you’d like another example or a different way to solve the quadratic (e.g., using the quadratic formula).