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QProblem
why did you divide by 2 first?
ASolution
Why we divided by 2 first
The original equation is
All three terms share a common factor of 2. Factoring that out gives
Why is this useful?
| Reason | What happens if we skip it? |
|---|---|
| Simplifies coefficients | The quadratic inside the parentheses has smaller numbers (1, –4, 3) that are easier to work with when we factor or use the quadratic formula. |
| Keeps the equation equivalent | Dividing every term by the same non‑zero number (or factoring it out) does not change the set of solutions; it merely rewrites the same equation. |
| Avoids arithmetic errors | Smaller numbers reduce the chance of mistakes when looking for factor pairs or computing the discriminant. |
| Makes the zero‑product step clear | After factoring out 2, the equation reduces to . Since the factor 2 can never be zero, the only way the product can be zero is if the quadratic factor is zero. This isolates the part we actually need to solve. |
So we divide by 2 (or factor it out) to simplify the problem while preserving all solutions. After solving the simpler quadratic, we obtain the same roots for the original equation.
If you’d like a visual explanation of this step and the overall factoring process, the video below walks through why and how we factor out the greatest common factor before solving a quadratic.