Study tips · 3 min read

10 Common Algebra Mistakes (With Numbers That Prove They’re Wrong)

Published October 3, 2026Written and reviewed by Mateuss M.

In short: A quick test exposes most algebra errors: substitute numbers. If (a + b)² = a² + b² were true, then with a = 3, b = 4 we would get 49 = 25 — but it is not.

1–3: Powers and roots

(a + b)² ≠ a² + b². At a = 3, b = 4: 49 vs 25. The correct expansion is a² + 2ab + b².
√(a + b) ≠ √a + √b. √(9 + 16) = 5, but √9 + √16 = 7.
−x² ≠ (−x)². At x = 3: −3² = -9 but (−3)² = 9.

4–6: Fractions and cancelling

Cancelling terms, not factors. (2 + 6)/2 ≠ 6: it is 4. Only common factors cancel.
a/b + c/d ≠ (a + c)/(b + d). 1/2 + 1/3 = 5/6, not 2/5.
Dividing by zero. Dividing both sides by an expression that might be 0 can lose a solution.

7–10: Signs and structure

Distributing a minus sign: −(x − 3) = −x + 3, not −x − 3.
Moving terms: when a term crosses the equals sign its sign changes.
Forgetting the order of operations: 2 + 3 × 4 = 14, not 20. See order of operations.
Not checking: always substitute your answer back into the original equation.

A check by substitution, built in to the linear solver

Answerx = −12

Step-by-step working 4 steps

1
Write the equation
```
3x + 5 = 2x − 7
```
2
Collect the x terms on the left
```
3x − 2x = −7 − 5
x = −12
```
Subtract the smaller x term and the constant from both sides, so x is alone on the left and numbers are on the right.
3
Divide by the coefficient of x
```
x = −12 ÷ 1 = −12
```
4
Check by substituting back
```
Left: 3(−12) + 5 = −31
Right: 2(−12) − 7 = −31
```
Both sides match, so the solution is correct.

A habit that helps

Test any new rule with small numbers before trusting it. For a list of correct identities see the algebra formulas cheat sheet, and practise with the linear equation solver.

Frequently asked questions

What is the most common algebra mistake?

Treating (a + b)² as a² + b², closely followed by sign errors when moving terms.

How can I catch my own errors?

Substitute your answer back and test rules with simple numbers.

Written and reviewed by Mateuss M.

Mateuss M. writes and reviews mathematical content for CalcSolver, focusing on online calculators, formulas, equations, and practical math tools. He reviews calculator functionality, calculation methods, formulas, examples, and explanations to help ensure that each tool is clear, useful, and easy to understand.

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