Tewtor.ai

Dividing by 2 is just a convenient algebraic step—it doesn’t change the solutions, it only makes the numbers easier to work with.

Why it’s allowed

An equation states that the left‑hand side (LHS) and the right‑hand side (RHS) are equal:

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

If we multiply (or divide) both sides by the same non‑zero number, the equality remains true because we’re performing the same operation on equal quantities. In other words,

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

Since 0 divided by 2 is still 0, the set of $x$ that satisfy the original equation is exactly the same as the set that satisfy the simplified one.

What the division accomplishes

• Smaller coefficients: $2, -8,$ and $6$ become $1, -4,$ and $3$.
• Easier factoring: With smaller numbers it’s quicker to spot a pair of integers whose product is the constant term (3) and whose sum is the linear coefficient (‑4).
• Less arithmetic: Fewer chances to make a slip‑up when you later expand or check your work.

If you preferred not to divide, you could still solve the quadratic by using the quadratic formula directly on the original coefficients $a=2,\;b=-8,\;c=6$. The result would be the same, but the arithmetic would involve larger numbers:

giving $x=1$ or $x=3$ again.

So the division by 2 is simply a strategic simplification that makes the factoring step more straightforward while preserving all solutions.