Common Mistakes
Learning Objectives
In this lesson, you will learn to:
- Identify common errors in factoring and solving quadratics
- Avoid sign errors when using the quadratic formula
- Remember to set equations equal to zero before factoring
- Check solutions by substitution
Practice Quiz
Find and correct the error in each problem. Click to reveal the answer.
Question 1: Error: To factor x^2 + 5x + 6, a student wrote (x + 2)(x + 4). Is this correct?
Answer: Incorrect - should be (x + 2)(x + 3)
Explanation: 2 x 4 = 8, not 6. The factors of 6 that add to 5 are 2 and 3.
Question 2: Error: Solving x^2 = 9, a student wrote x = 3 only. What's wrong?
Answer: Missing x = -3
Explanation: Both 3 and -3 square to give 9. Always include both the positive and negative square roots.
Question 3: Error: For x^2 + 4x = 12, a student factored x(x + 4) = 12 and concluded x = 12 or x + 4 = 12. What's wrong?
Answer: Must set equal to zero first
Explanation: Correct method: x^2 + 4x - 12 = 0, then factor (x+6)(x-2) = 0. x = -6 or x = 2.
Question 4: Error: Using quadratic formula for x^2 - 6x + 5 = 0, a student wrote b = 6. What's wrong?
Answer: b = -6, not 6
Explanation: The coefficient of x is -6, so b = -6. In the formula, -b would then equal -(-6) = 6.
Question 5: Error: (x + 3)^2 = x^2 + 9. Is this correct?
Answer: Incorrect - missing the middle term
Explanation: (x + 3)^2 = x^2 + 6x + 9. Remember: (a + b)^2 = a^2 + 2ab + b^2.
Question 6: Error: sqrt(x^2 + 16) = x + 4. Is this correct?
Answer: Incorrect
Explanation: You cannot "distribute" a square root. sqrt(x^2 + 16) does not simplify to x + 4. For example, sqrt(9 + 16) = sqrt(25) = 5, not 3 + 4 = 7.
Question 7: Error: For 2x^2 - 8 = 0, a student divided by 2 to get x^2 - 4 = 0, then said x = 4. What's wrong?
Answer: x = +/-2, not 4
Explanation: x^2 = 4 means x = +2 or -2 (square root of 4). The student confused x^2 = 4 with x = 4.
Question 8: Error: x^2 - x - 6 = (x - 3)(x - 2). Is this correct?
Answer: Incorrect - should be (x - 3)(x + 2)
Explanation: (-3)(+2) = -6 and -3 + 2 = -1. Check: (x-3)(x+2) = x^2 - x - 6.
Question 9: Error: In the discriminant b^2 - 4ac for x^2 + 2x + 5 = 0, a student calculated 4 - 20 = 16. What's wrong?
Answer: Should be 4 - 20 = -16
Explanation: b^2 - 4ac = (2)^2 - 4(1)(5) = 4 - 20 = -16. Negative discriminant means no real solutions.
Question 10: Error: For x^2 - 4x = 0, a student divided both sides by x to get x - 4 = 0, so x = 4. What's missing?
Answer: Missing solution x = 0
Explanation: Factor instead: x(x - 4) = 0. Then x = 0 or x = 4. Dividing by x loses the solution x = 0.
Next Steps
- Always check solutions by substituting back
- Watch for sign errors in the quadratic formula
- Take the unit quiz when ready