In this article
- The quick test that catches most mistakes
- 1. Squaring a bracket term by term
- 2. Losing a minus sign
- 3. Cancelling terms instead of factors
- 4. Splitting a fraction the wrong way
- 5. Treating roots like brackets
- 6. Dividing by something that could be zero
- 7. Forgetting to check the answer
- A study method that works: learn from worked mistakes
- Frequently asked questions
Ask any maths teacher where students lose marks in algebra and you will hear the same answer. It is rarely the hard topics. It is the small slips that happen again and again: a lost minus sign, a bracket squared the wrong way, a fraction cancelled when it should not be.
The good news is that these mistakes follow patterns. Once you know them, you can check for them in seconds. This guide covers the most common ones, with a quick test for each.
Key facts
- Most errors are habits, not gaps in understanding. The same few mistakes appear across all levels of algebra.
- Plugging in a number is the fastest way to test any step. If the two sides give different answers, the step is wrong.
- Studying worked mistakes is one of three strategies recommended by the US Department of Education’s What Works Clearinghouse for teaching algebra.
The quick test that catches most mistakes
Before looking at specific errors, learn one trick. If you are not sure a step is correct, replace the letter with a simple number, such as 2 or 3, and work out both sides. If the answers do not match, the step is wrong.
Example: is (x + 3)² the same as x² + 9? Try x = 2. The left side is (2 + 3)² = 25. The right side is 4 + 9 = 13. They do not match, so the step is wrong.
1. Squaring a bracket term by term
This is probably the most common algebra error of all. Squaring a sum is not the same as squaring each part.
| Wrong | Right |
|---|---|
| (a + b)² = a² + b² | (a + b)² = a² + 2ab + b² |
| (x − 5)² = x² − 25 | (x − 5)² = x² − 10x + 25 |
Remember that (a + b)² means (a + b)(a + b). Multiply it out and the middle term appears.
2. Losing a minus sign
Minus signs cause more lost marks than any single topic. Three situations cause most of the trouble.
- Subtracting a bracket. 10 − (x − 4) is 10 − x + 4, which is 14 − x. The minus applies to everything inside the bracket.
- Squaring a negative. (−3)² is 9, but −3² is −9. The bracket makes the difference.
- Moving terms across the equals sign. If x + 7 = 2, then x = 2 − 7 = −5. Do the same thing to both sides, and write each step.
3. Cancelling terms instead of factors
You can only cancel things that are multiplied, not things that are added.
| Wrong | Right |
|---|---|
| (x + 6) ÷ 6 = x | (x + 6) ÷ 6 = x/6 + 1 |
| (2x + 4) ÷ 2 = x + 4 | (2x + 4) ÷ 2 = x + 2 |
Try the quick test: with x = 6, (6 + 6) ÷ 6 = 2, not 6. Factorising first, such as 2x + 4 = 2(x + 2), shows you what can safely be cancelled.
4. Splitting a fraction the wrong way
You can split the top of a fraction, but not the bottom. So (a + b)/c = a/c + b/c is correct. But c/(a + b) is not c/a + c/b. Try a = b = c = 1: the left side is 1/2, while the right side is 2.
5. Treating roots like brackets
Just as squares do not split over addition, neither do square roots. The square root of (9 + 16) is the square root of 25, which is 5. It is not 3 + 4 = 7. Roots only split over multiplication, so √(4 × 9) = √4 × √9 = 6.
6. Dividing by something that could be zero
If x² = 3x, it is tempting to divide both sides by x and get x = 3. But that loses an answer. x = 0 also works. Instead, move everything to one side and factorise: x² − 3x = 0, so x(x − 3) = 0, which gives x = 0 or x = 3.
7. Forgetting to check the answer
When you solve an equation, put your answer back into the original question. This catches nearly every slip above. It takes less than a minute and is one of the easiest ways to protect your marks in an exam.
A study method that works: learn from worked mistakes
The What Works Clearinghouse, part of the US Department of Education, reviewed the research on teaching algebra. Its practice guide makes three recommendations. The first is to use solved problems, including ones with errors, and ask students to explain what went right or wrong.
You can do this on your own:
- Keep a list of the mistakes you made in homework and tests.
- For each one, write the wrong step and the correct step side by side.
- Write one sentence explaining why the wrong step is wrong.
- Before an exam, read the list and check your answers for those exact errors.
For more examples, the free common algebra errors page from Lamar University’s online maths notes lists many of these mistakes with worked examples, and its common math errors section covers other topics too.
Frequently asked questions
What is the most common algebra mistake?
Squaring a bracket term by term, such as writing (a + b)² as a² + b², is one of the most common. Sign errors with brackets and negative numbers are close behind.
How can I stop making careless mistakes in maths?
Write every step, check each answer by putting it back into the original equation, and test doubtful steps by replacing the letter with a simple number.
Is it better to practise more problems or study mistakes?
Both help. Research reviewed by the What Works Clearinghouse supports studying solved problems, including incorrect ones, alongside regular practice.
For a practical use of algebra in everyday life, see how watts, joules and your electricity bill are connected, or browse the Mathematics section.
