A second method is a tool for checking an answer. Used that way, it makes you more confident and adds little to what you must memorise.
This page is part of the SPM Additional Mathematics guide. A check is also how you catch a slip early, as in recovering after an early mistake.
What does the student usually say?
“My teacher showed two ways, and now I am not sure which to use.” Or: “I already know a method. Learning a second one feels like repeating myself.”
Two methods, one answer
The question, invented for this page: find the minimum value of y = x² − 6x + 4.
Method 1, completing the square. Write y = (x − 3)² − 9 + 4 = (x − 3)² − 5. A squared term is never negative, so the smallest value is −5 when x = 3.
Method 2, differentiation. dy/dx = 2x − 6. Set it to 0, so x = 3. Then y = 9 − 18 + 4 = −5.
Both give a minimum of −5 at x = 3. You can trust the answer because two separate routes reached it, and you can see why it is right, not only how to write it.
When does a second method help?
| Situation | Second method? | Why |
|---|---|---|
| Quadratic minimum or maximum | Yes | Completing the square and calculus check each other |
| Long differentiation with a fraction | Sometimes | Substituting a value is faster than a full second method |
| A one-step substitution | No | The original line is the check |
| A high-mark question with spare time | Yes | The cost of a slip is larger |
What can I do myself?
- Choose one topic, such as quadratic functions, and learn a second method for it only.
- Solve five questions by your main method, then check each with the second.
- When the answers disagree, find the line where they split and note the slip in the mistake log and paper-error review.
- Drop the second method for a question once the main method is reliable.
When might one-to-one tuition help?
If you keep getting two different answers and cannot tell which is wrong, a teacher can compare the two sets of working with you and show which line diverges. That is hard to do alone, because you wrote both.
Bring a question you solved twice with different answers to the one-hour trial class (from RM50). Read about online one-to-one Additional Mathematics tuition for how lessons work.