MathSetuMost marks lost in algebra are not lost because the student does not understand the chapter. They are lost on small slips: a missing sign, a wrongly cancelled term, a squared bracket handled in one step. The good news is that there is a quick check that catches nearly all of them. We will meet it first, then use it on five classic mistakes.
The number test
An identity is true for every value of the letters. So if your two sides disagree for even one value, your step is wrong. Pick an easy number, such as x = 2, put it into both sides, and compare. It takes ten seconds and you can do it in the margin of your rough work.
Mistake 1: (a + b)² = a² + b²
This is the most famous one. Test it with a = 2 and b = 3. The left side is (2 + 3)² = 25. The right side is 4 + 9 = 13. They differ, so the rule is false. The correct identity is (a + b)² = a² + 2ab + b². Check: 4 + 12 + 9 = 25. The missing piece, 2ab, comes from the two rectangles in the square diagram. For example, (x + 3)² = x² + 6x + 9.
Mistake 2: the sign in the middle term
(a − b)² = a² − 2ab + b². Only the middle term is negative; the last term b² is always positive. So (2x − 5)² = 4x² − 20x + 25. Test with x = 3: the left side is (6 − 5)² = 1, and the right side is 36 − 60 + 25 = 1. It matches.
Mistake 3: subtracting a bracket
When a minus sign sits in front of a bracket, it changes the sign of every term inside. Take (3x + 5) − (x − 4). Many students write 3x + 5 − x − 4 and get 2x + 1. The correct step is 3x + 5 − x + 4, which gives 2x + 9. Test with x = 1: the original is 8 − (−3) = 11, and 2x + 9 = 11. The wrong answer 2x + 1 gives 3, so the test exposes it at once.
Mistake 4: cancelling a term instead of a factor
You may cancel a common factor, never a single term. In (x + 6) / 6 you cannot cancel the 6 and write x. Test with x = 6: the fraction is 12 / 6 = 2, but x is 6. The correct simplification is x/6 + 1, which gives 1 + 1 = 2. Compare this with (6x) / 6 = x, where 6 is a factor of the whole top line and cancelling is allowed.
Mistake 5: forgetting to check a factorisation
To factorise x² − 5x + 6, we look for two numbers with product 6 and sum −5. These are −2 and −3, so x² − 5x + 6 = (x − 2)(x − 3). Always multiply back: x² − 3x − 2x + 6 = x² − 5x + 6. In Class 9 you can also use the factor theorem: put x = 2 into the polynomial and you get 4 − 10 + 6 = 0, so (x − 2) is a factor. A zero remainder is another built-in check.
Make it a habit
After you finish a long expansion or simplification, spend ten seconds substituting x = 1 or x = 2 on the first line and the last line. If the values match, you can be fairly confident. If not, hunt for the sign or the missing term. Avoid x = 0 and x = 1 when a term is multiplied by x, because those values can hide an error. Choose something like 2 or 3 as well. This habit is part of the revision routine in our post on how to study a Maths chapter.
Try it yourself
- Expand (x + 4)² and test your answer with x = 2.
- Simplify (5x − 2) − (2x − 7) and test with x = 3.
- Is (x + 10) / 5 equal to x + 2? Test with x = 5, then simplify it correctly.
- Factorise x² − 7x + 12 and multiply back to check.
Answers: 1) x² + 8x + 16 (at x = 2, both sides give 36). 2) 3x + 5 (at x = 3, both give 14). 3) No. At x = 5 the left side is 3 but x + 2 is 7. The correct form is x/5 + 2. 4) (x − 3)(x − 4). For more questions of this kind, see the Class 9 chapters and generate a fresh set in the Practice Generator.