Quadratic Formula Error Analysis
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Quadratic Formula Error Analysis
Part 1: Formula Reminder and Error Analysis
Learning target: I can solve quadratic equations using the quadratic formula and explain mistakes in a solution.
Quadratic formula: For an equation in standard form, ax² + bx + c = 0, use x = (-b ± √(b² - 4ac)) / (2a).
Before using the formula, check: (1) Move every term to one side so the equation is ax² + bx + c = 0. (2) Identify a, b, and c, including their signs. (3) Substitute carefully. (4) Simplify both answers when possible.
For each problem, circle or describe the first incorrect step, explain the error, correct the work, and solve the related equation independently.
Incorrect work: a = 2, b = −5, c = −3
x = (−5 ± √(25 + 24)) / 4
x = (−5 ± 7) / 4, so x = 1/2 or x = −3
Error and correction: ________________________________________________
Related equation: Solve x² − 7x + 10 = 0.
Incorrect work: a = 1, b = 6, c = 5
b² = 6² = 12
x = (−6 ± √(12 − 20)) / 2 = (−6 ± √−8) / 2
Error and correction: ________________________________________________
Related equation: Solve x² + 4x − 5 = 0.
Incorrect work: a = 3, b = 2, c = −1
b² − 4ac = 4 − 4(3)(−1) = 4 − 12 = −8
x = (−2 ± √−8) / 6
Error and correction: ________________________________________________
Related equation: Solve 2x² − 5x − 3 = 0.
Incorrect work: a = 2, b = 7, c = 3
x = (−7 ± √(49 − 24)) / 2
x = (−7 ± 5) / 2, so x = −1 or x = −6
Error and correction: ________________________________________________
Related equation: Solve 3x² − 8x + 4 = 0.
Incorrect work: a = 1, b = −4, c = 1
x = (4 + √(16 − 4)) / 2 = (4 + √12) / 2 = 2 + √3
Error and correction: ________________________________________________
Related equation: Solve x² + 2x − 8 = 0.
Incorrect work: a = 1, b = 0, c = 2
x = (0 ± √(0 − 8)) / 2 = ±i√2
Error and correction: ________________________________________________
Related equation: Solve 2x² − 3x − 2 = 0.
Part 2: Mixed Practice
Use the quadratic formula. Give exact answers. If there are no real solutions, state that and write the complex solutions if appropriate.
Answer Key
1. Error: −b was treated as −5 instead of 5. Correct solutions: x = 3 and x = −1/2. Related: x = 5, 2.
2. Error: 6² = 36, not 12. Correct solutions: x = −1, −5. Related: x = 1, −5.
3. Error: −4ac = −4(3)(−1) = +12, not −12. Correct solutions: x = 1/3, −1. Related: x = 3, −1/2.
4. Error: The denominator is 2a = 4, not 2. Correct solutions: x = −1/2, −3. Related: x = 2, 2/3.
5. Error: The ± was omitted. Correct solutions: x = 2 + √3 and x = 2 − √3. Related: x = 2, −4.
6. Error: The equation was not written in standard form. Correct form: x² − 3x + 2 = 0. Correct solutions: x = 1, 2. Related: x = 2, −1/2.
Mixed practice: 7. x = 4, 5. 8. x = 1 ± √6/2. 9. No real solutions; complex solutions are x = (−1 ± i√14) / 3. 10. x = −3 ± 2√2.
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