Common Mistakes | Algebra II Completion

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Understanding common mistakes is just as important as learning correct methods. By recognizing these error patterns, you can check your work more effectively and avoid losing points on tests.

Categories of Common Errors

Sign errors: Mistakes with positive and negative numbers
Order of operations: Performing operations in the wrong sequence
Distribution errors: Incorrectly applying the distributive property
Conceptual errors: Misunderstanding fundamental rules
Notation errors: Misreading or miswriting mathematical symbols

Complex Numbers: Common Errors

Error: Treating i as a variable instead of sqrt(-1)

Example: Writing i^2 = i * i = i^2 instead of i^2 = -1

Correction: Always remember that i^2 = -1, i^3 = -i, i^4 = 1, then the cycle repeats.

Error: Forgetting to distribute the negative when subtracting complex numbers

Example: (3 + 2i) - (1 - 4i) = 3 + 2i - 1 - 4i = 2 - 2i (wrong)

Correction: (3 + 2i) - (1 - 4i) = 3 + 2i - 1 + 4i = 2 + 6i (correct)

Polynomials: Common Errors

Error: Incorrectly applying exponent rules

Example: (x + 2)^2 = x^2 + 4 (wrong)

Correction: (x + 2)^2 = x^2 + 4x + 4 (correct) - Remember to FOIL or use the formula a^2 + 2ab + b^2

Error: Dropping terms during polynomial division

Example: When dividing x^3 + 1 by (x + 1), forgetting to include 0x^2 + 0x as placeholders

Correction: Write as x^3 + 0x^2 + 0x + 1 before dividing to keep track of all terms

Equations: Common Errors

Error: Not checking for extraneous solutions

Example: When solving sqrt(x + 3) = x - 1, accepting x = -2 as a solution

Correction: Always substitute back: sqrt(-2 + 3) = sqrt(1) = 1, but -2 - 1 = -3. Since 1 ≠ -3, reject this solution.

Examples

Identify the error in each problem, then see the correct solution.

Example 1: Find the Error

Problem: Simplify (2 + 3i)^2

Incorrect Solution: (2 + 3i)^2 = 4 + 9i^2 = 4 + 9(-1) = 4 - 9 = -5

Error: Failed to include the middle term when squaring

Correct Solution: (2 + 3i)^2 = 4 + 12i + 9i^2 = 4 + 12i - 9 = -5 + 12i

Example 2: Find the Error

Problem: Factor x^2 - 9

Incorrect Solution: x^2 - 9 = (x - 3)^2

Error: Confused difference of squares with perfect square trinomial

Correct Solution: x^2 - 9 = (x + 3)(x - 3) (difference of squares pattern)

Example 3: Find the Error

Problem: Solve x^2 = 16

Incorrect Solution: x = 4

Error: Forgot the negative solution

Correct Solution: x = 4 or x = -4 (both values satisfy the equation)

Practice

For each problem, identify the error and provide the correct solution.

Problem 1: Find the error: sqrt(x^2) = x for all real numbers

Problem 2: Find the error: (a + b)^3 = a^3 + b^3

Problem 3: Find the error: When solving x^2 - 5x + 6 = 0, a student wrote x(x - 5) + 6 = 0, so x = 0 or x = 5 or x = -6

Problem 4: Find the error: i^100 = i

Problem 5: Find the error: The zeros of f(x) = x^3 - 4x are x = 0 and x = 4

Problem 6: Find the error: (3/x) + (2/x) = 5/2x

Problem 7: Find the error: If f(x) = x^2 - 4, then f(a + h) = a^2 + h^2 - 4

Problem 8: Find the error: The vertex of y = (x - 3)^2 + 5 is at (-3, 5)

Problem 9: Find the error: To solve |x - 2| = 5, a student wrote x - 2 = 5, so x = 7

Problem 10: Find the error: When using the quadratic formula on 2x^2 + 3x - 5 = 0, a student wrote x = (-3 ± sqrt(9 - 40)) / 4

Problem 11: Find the error: log(x + y) = log(x) + log(y)

Problem 12: Find the error: When dividing (x^3 - 8) by (x - 2), a student started synthetic division with coefficients 1, -8

Check Your Understanding

Reflect on strategies to prevent common mistakes.

What is a quick way to check if you correctly squared a binomial?
Why is it important to use placeholders for missing terms in polynomial division?
How can you avoid sign errors when subtracting complex numbers?
What should you always do after solving an equation that involved squaring both sides?

Next Steps

Create an error log to track mistakes you commonly make
Review the errors covered here before taking tests
Double-check your work by substituting answers back into original equations
Take the Unit Quiz to test your understanding of the entire unit