Study skills · error review

Turn a careless mistake into a check you can repeat

“Be more careful” is hard to act on. A useful correction names the exact decision that went wrong and gives you a short check to use next time. These three examples show how.

By Sterling Education Studio · 3 October 2026 · Original practice examples

The key idea

After a mistake, identify the first incorrect step, explain why it was invalid and write a check tied to that operation. Then solve a different question that needs the same check. An error log is useful when it changes what you do on the next attempt.

1. Dividing by an expression can lose a solution

Solve x(x − 3) = 2x. Dividing both sides by x gives x − 3 = 2 and hence x = 5. But that division assumes x ≠ 0.

A safer route keeps both possibilities:

x(x − 3) − 2x = 0
x(x − 5) = 0
x = 0 or x = 5

Substituting x = 0 into the original equation gives 0 = 0, so it is a valid solution.

Repeatable check: before dividing by an expression containing x, ask when it is zero. Either handle that case separately or rearrange and factorise instead.

Retry: solve x(x + 1) = 4x.

Check the retry

x(x − 3) = 0, so x = 0 or x = 3. Dividing by x without considering zero would lose x = 0 again.

2. An unknown denominator has an unknown sign

Solve 1/(x − 1) > 2. Multiplying by x − 1 and keeping the same inequality sign assumes x − 1 is positive. That is not true for every allowed x.

Bring everything to one side:

1/(x − 1) − 2 > 0
(3 − 2x)/(x − 1) > 0

The critical values are x = 1, where the expression is undefined, and x = 3/2, where the numerator is zero.

For x < 1, the numerator is positive and the denominator negative: the fraction is negative.
For 1 < x < 3/2, both are positive: the fraction is positive.
For x > 3/2, the numerator is negative and the denominator positive: the fraction is negative.

Answer: 1 < x < 3/2. Both endpoints are excluded, for different reasons.

Repeatable check: before multiplying an inequality by an expression, establish its sign. For a rational expression, a sign analysis also makes excluded values visible.

Retry: solve 1/(x + 2) > 1.

Check the retry

Rearrange to (−x − 1)/(x + 2) > 0. The critical values are −2 and −1. The fraction is positive only for −2 < x < −1.

3. Squaring can introduce an extra candidate

Solve √x = x − 2. The left-hand side is non-negative, so a solution must have x ≥ 2.

Squaring gives:

x = (x − 2)²
x² − 5x + 4 = 0
(x − 1)(x − 4) = 0

The candidates are 1 and 4. In the original equation, x = 1 gives 1 = −1, which is false. At x = 4, both sides equal 2. The only solution is x = 4.

Repeatable check: after an operation that may not be reversible, substitute candidate solutions into the original equation and check the original restrictions.

Make a short error log that changes your working

Choose one recent question and write four lines:

First incorrect step: “I divided by x.”
Why it failed: “That assumed x ≠ 0, but zero satisfied the original equation.”
Next-time check: “Check the zero case before dividing by a variable expression.”
Evidence it is fixed: “I solved a fresh question and kept both valid solutions without looking at the correction.”

If you still cannot explain why a step failed, return to the idea behind it before doing more timed questions. If you can explain it but skip the check under time pressure, practise placing the check directly beside the risky operation.

Use your checking time where it can catch something

Before a paper, choose two recurring errors from your own work. During the paper, mark the operations where those errors tend to happen. If time remains, revisit those points, then check whether each final answer satisfies the question’s conditions.

For inverse-function questions, for example, a domain check is more targeted than rereading every line. The inverse-functions guide shows how that check determines the square-root branch.

A final answer that looks reasonable can still come from invalid working. Aim to check both the result and the step most likely to have gone wrong.

Keep learning

These are original teaching examples, not past-paper questions or an official mark scheme. Use the 2026 H2 Mathematics 9758 syllabus guide to check the wider topic requirements for that examination year.

Planning what comes next? Read the promos recovery plan or JC2 preparation guide.