Tewtor.ai

When we solve a quadratic equation, any operation that multiplies every term by the same non‑zero number leaves the set of solutions unchanged.

In the equation

the coefficients 2, –8, and 6 all share a common factor of 2. Dividing the entire equation by that greatest common factor (GCF) gives a simpler, “equivalent” equation:

Why this helps

1. Smaller numbers – Working with 1, –4, and 3 is easier than with 2, –8, and 6 when we look for factors or apply the quadratic formula.
2. Preserves solutions – Because we divided both sides of the equation by the same non‑zero constant (2), any value of $x$ that satisfies the original equation also satisfies the reduced one, and vice‑versa.
3. No loss of information – The step does not discard any possible roots; it merely rewrites the same relationship in a more convenient form.

After the division, we can factor the quadratic:

which directly yields the solutions $x=1$ and $x=3$. If we had tried to factor the original form $2x^{2}-8x+6$ we would need to find two numbers whose product is $2\times6=12$ and whose sum is $-8$; the numbers $-6$ and $-2$ work, but the algebra is a little messier. Dividing by the GCF streamlines the process.

So, dividing by 2 first is simply a convenient algebraic simplification that keeps the equation equivalent while making the subsequent steps easier.

Do you want a visual demonstration of how factoring works after the GCF is removed? I can provide an interactive quadratic‑explorer or another video if that would help. Let me know!