Two methods are equally valid when both are correct and both answer the question. They differ in speed, risk of error, how easily you can check them and how well they suit the question’s wording. A method is not better because it looks advanced.
This page helps if your results are already strong and you want to sharpen choices rather than learn more content. It avoids one claim completely: no method will move you to a particular grade.
What criteria compare methods fairly?
Use four, and score each method briefly, in words.
| Criterion | Question to ask |
|---|---|
| Speed | How many lines does it need for this question? |
| Error risk | Where is this method most likely to slip (signs, squares, rounding)? |
| Checkability | Can I test the answer quickly by substitution? |
| Fit | Does the question say “exact”, “to 2 decimal places”, “hence”, or name a method? |
These criteria are about you and the question, not about what an examiner might like.
Worked example: one quadratic, three methods
Solve x² + 4x − 5 = 0.
Method A, factorising. Find two numbers that multiply to −5 and add to 4: 5 and −1. So (x + 5)(x − 1) = 0, giving x = −5 or x = 1.
Method B, the quadratic formula. a = 1, b = 4, c = −5. The discriminant is b² − 4ac = 16 + 20 = 36. So x = (−4 ± 6) ÷ 2, giving x = 1 or x = −5.
Method C, completing the square. x² + 4x = 5, so (x + 2)² = 9, x + 2 = ±3, giving x = 1 or x = −5.
All three give x = 1 and x = −5. Check by substitution: 1 + 4 − 5 = 0, and 25 − 20 − 5 = 0.
| A: Factorise | B: Formula | C: Complete the square | |
|---|---|---|---|
| Speed | 2 lines | 3 to 4 lines | 3 lines |
| Error risk | Low if numbers are small | Sign slip with −4ac | Halving b, carrying the constant |
| Checkability | Easy, expand back | Easy, substitute | Easy, substitute |
| Fit | Needs integer roots | Works for any; good for rounding | Useful when a turning point is needed |
For this question, A is fastest. It is not the “better” method for every quadratic.
When the question changes the choice
Try x² + 3x − 5 = 0, answer to 2 decimal places.
Factorising fails, because no integer pair multiplies to −5 and adds to 3. The formula works: discriminant = 9 + 20 = 29, √29 ≈ 5.385, so x = (−3 + 5.385) ÷ 2 ≈ 1.19 or x = (−3 − 5.385) ÷ 2 ≈ −4.19.
Notice what drove the choice: the wording (“2 decimal places”) and the numbers, not a ranking.
The mistake to avoid
The trap is believing there is one “correct” method and that using another shows weakness. The opposite trap is switching methods halfway through a question because the first felt slow. That usually costs more time than finishing.
A third trap is treating a comparison as a prediction. “I use the formula, so I must be at a high level” is not a valid inference. The evidence you can collect is narrower: on comparable tasks, which method gave fewer errors for you?
Check yourself
1. Solve x² − 7x + 12 = 0. Which method is quickest, and why?
Show answer
Factorise: two numbers multiplying to 12 and adding to −7 are −3 and −4, so (x − 3)(x − 4) = 0 and x = 3 or x = 4. Factorising is quickest because the numbers are small integers. Check: 9 − 21 + 12 = 0 and 16 − 28 + 12 = 0.
2. Why is the quadratic formula the natural choice for x² + 3x − 5 = 0 to 2 decimal places?
Show answer
The roots are not integers, so factorising does not work, and the question asks for decimals, which the formula gives directly: 1.19 and −4.19.
3. A classmate says, “I use the formula for everything, so I do not need to learn factorising.” Give one fair point for and one against.
Show answer
For: the formula always works, so there is no need to decide. Against: it is slower and risks sign slips on simple questions, and factorising lets you check quickly. The fair conclusion is that knowing both gives a choice and a self-check.
Where this leads next
Record which method you used and whether it was right, on tasks of similar difficulty, in the evidence-of-progress tracker. That gives you like-for-like evidence instead of a feeling. Related pages: identifying the final reasoning gap and checking an answer for relevance.
You can run this comparison alone. A one-to-one teacher becomes worth considering when you want someone to time both methods with you and watch where each one slips.
In online one-to-one Mathematics tuition, that is a normal part of a lesson. The paid trial is one way to see it in practice, and student learning routes list more.