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Why we divide the whole equation by 2 first

The original quadratic is

All three coefficients (2, –8, 6) share a common factor of 2. Dividing every term by that common factor gives an equivalent equation that is algebraically simpler:

What “equivalent” means

When we divide every term by the same non‑zero number, we are multiplying both sides of the equation by the same factor (here, $\tfrac12$). This does not change the set of solutions; any $x$ that satisfies the original equation will also satisfy the reduced one, and vice‑versa.

Why it helps

1. Easier factoring – With a leading coefficient of 1, we can look for two numbers that multiply to the constant term (3) and add to the linear coefficient (–4). That’s straightforward: –1 and –3, giving $(x-1)(x-3)=0$.

2. Simpler quadratic formula – If we kept the original coefficients, the formula would be

   $$x = \frac{-(-8) \pm \sqrt{(-8)^{2} - 4(2)(6)}}{2(2)} ,$$

   which simplifies to the same result after a few extra arithmetic steps. Dividing first reduces the arithmetic load.

3. Avoids mistakes – Working with smaller numbers lessens the chance of arithmetic errors, especially when you’re practicing the method.

Important caution

You must divide every term (including the constant) by the same factor. Dividing only part of the equation (e.g., just the $2x^{2}$ term) would break the equality and lead to an incorrect solution.

If you’d like a visual explanation of how dividing by a common factor works in solving quadratics, the video below covers it in detail.

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