Week 7 — Quiz (auto-graded) · Quadratic Equations

Course: College Algebra (MATH 120) · Silver Oak University (fictional sample) · Prof. Calloway
Objective tested: Objective 6 — factoring & zero-product property; square root property; completing the square; quadratic formula; discriminant.
Points: 10 (1 each) · Assignment group: Quizzes (15% of grade) · Due: end of Module 7.

This is the human-readable quiz with its vetted answer key and feedback. The import-ready Classic QTI is in F-quiz-week-07-qti.xml. AI is not permitted on quizzes (course AI policy). Every numeric answer below is pre-computed and independently re-verified (Python w07_verify.py, PASS).

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Blueprint

#	Type	Concept	Objective
1	Multiple choice	Factoring & zero-product: x²−5x+6=0	6
2	Multiple choice	Zero-product property: (x−4)(x+1)=0	6
3	Multiple choice	Square root property: x²=49	6
4	Multiple choice	Square root property (shifted): (x−3)²=16	6
5	Multiple choice	Factoring: x²+2x−15=0	6
6	Multiple choice	Quadratic formula: x²−4x+1=0	6
7	Multiple choice	Discriminant (negative): x²+2x+5=0	6
8	Matching	Discriminant sign ↔ number of real roots	6
9	Multiple choice	Completing the square: x²+6x+5=0	6
10	Multiple choice	Application: x(x+3)=40	6

No trick questions; distractors target the Week 7 misconceptions named in the lecture outline.

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Questions, key, and feedback

Q1 (MC). Solve by factoring: x² − 5x + 6 = 0

• A. x = 2 or x = 3 ✅
• B. x = −2 or x = −3
• C. x = 2 or x = −3
• D. x = −2 or x = 3

Feedback: Factor by finding two numbers multiplying to +6 and adding to −5: that's −2 and −3. So (x−2)(x−3) = 0, giving x = 2 or x = 3. (B negates both unnecessarily; C/D mix signs.)

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Q2 (MC). Solve: (x − 4)(x + 1) = 0

• A. x = 4 or x = −1 ✅
• B. x = −4 or x = 1
• C. x = 4 or x = 1
• D. x = −4 or x = −1

Feedback: Zero-product property — set each factor to zero: x − 4 = 0 → x = 4; x + 1 = 0 → x = −1. (B flips both signs; C/D partially flip.)

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Q3 (MC). Use the square root property to solve: x² = 49

• A. x = ±7 ✅
• B. x = 7 only
• C. x = ±√49 = ±7.07 (approx)
• D. x = ±24.5

Feedback: The square root property gives x = ±√49 = ±7 exactly. (B drops the − solution; C miscomputes √49; D is ±49/2.) Both x = 7 and x = −7 satisfy the equation.

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Q4 (MC). Solve using the square root property: (x − 3)² = 16

• A. x = 7 or x = −1 ✅
• B. x = 7 only
• C. x = 5 or x = 1
• D. x = 19 or x = −13

Feedback: Take ±√16 = ±4. Then x − 3 = 4 → x = 7, and x − 3 = −4 → x = −1. (B drops the negative case; C takes ±4 but adds to 3 wrong; D squares 4 again.) Check: (7−3)² = 16 ✓, (−1−3)² = 16 ✓.

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Q5 (MC). Solve by factoring: x² + 2x − 15 = 0

• A. x = 3 or x = −5 ✅
• B. x = −3 or x = 5
• C. x = 3 or x = 5
• D. x = 15 or x = −1

Feedback: Two numbers multiplying to −15 and adding to +2: +5 and −3. So (x+5)(x−3)=0 → x = −5 or x = 3. (B reverses the signs; C gives both positive; D uses the constant and b only.)

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Q6 (MC). Use the quadratic formula to solve: x² − 4x + 1 = 0

• A. x = 2 ± √3 ✅
• B. x = −2 ± √3
• C. x = 2 ± √15
• D. x = 4 ± √3

Feedback: a=1, b=−4, c=1. Discriminant = 16−4 = 12. x = (−(−4) ± √12)/(2) = (4 ± 2√3)/2 = 2 ± √3. (B uses b instead of −b; C uses wrong discriminant 16−1=15; D forgets to divide by 2a.)

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Q7 (MC). Compute the discriminant of x² + 2x + 5 = 0 and interpret it.

• A. Discriminant = −16; no real solutions ✅
• B. Discriminant = 24; two real solutions
• C. Discriminant = 16; two real solutions
• D. Discriminant = 0; one repeated root

Feedback: a=1, b=2, c=5. Discriminant = b²−4ac = 4−20 = −16. Since it is negative, there are no real solutions. (B computes 4+20=24; C takes |4−20|=16; D misidentifies as zero.)

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Q8 (Matching). Match each discriminant condition to the correct number and type of real solutions.

Discriminant condition	Number and type of real solutions
b²−4ac > 0 (positive)	Two distinct real solutions
b²−4ac = 0 (zero)	Exactly one repeated real solution
b²−4ac < 0 (negative)	No real solutions

Feedback: The discriminant is the "preview screen." Positive → graph crosses the x-axis twice (two solutions). Zero → graph just touches the x-axis (one repeated solution). Negative → graph never reaches the x-axis (no real solutions).

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Q9 (MC). Solve by completing the square: x² + 6x + 5 = 0

• A. x = −1 or x = −5 ✅
• B. x = 1 or x = 5
• C. x = −1 or x = 5
• D. x = 1 or x = −5

Feedback: x²+6x = −5 → add (6/2)²=9 to both sides → (x+3)²=4 → x+3=±2 → x=−1 or x=−5. (B takes both roots positive (+1 and +5); C and D each flip the sign of exactly one root, the classic ± slip when undoing (x+3)²=4.)

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Q10 (MC). A rectangle has length x + 3 and width x. Its area is 40 square feet. Which value of x makes physical sense?

• A. x = 5 ✅
• B. x = −8
• C. x = 5 or x = −8 (both)
• D. x = 4

Feedback: Area: x(x+3) = 40 → x²+3x−40=0. Discriminant = 9+160=169 → x=(−3±13)/2 → x=5 or x=−8. Since x is a width, x must be positive → x = 5. (B takes the negative root; C doesn't eliminate the non-physical root; D is an arithmetic slip.)

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Answer key (quick reference)

Q	Answer
1	A (x=2 or x=3)
2	A (x=4 or x=−1)
3	A (x=±7)
4	A (x=7 or x=−1)
5	A (x=3 or x=−5)
6	A (x=2±√3)
7	A (discriminant=−16, no real solutions)
8	positive→two / zero→one repeated / negative→none
9	A (x=−1 or x=−5)
10	A (x=5)

Quality gate (self-checked, computer-verified): each single-answer item has exactly one correct option; the matching item pairs 1:1 (three pairs). Arithmetic pre-computed and independently re-verified (w07_verify.py, PASS): Q1 x²−5x+6={2,3}; Q2 (x−4)(x+1)={4,−1}; Q3 x²=49→±7; Q4 (x−3)²=16→{7,−1}; Q5 x²+2x−15={3,−5}; Q6 discriminant=12, x=2±√3; Q7 discriminant=−16; Q8 (definitional); Q9 completing square→{−1,−5}; Q10 x²+3x−40=0→x=5 (positive). All checks PASS. QTI parse confirmation: F-quiz-week-07-qti.xml parses with 10 items.

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Item-bank entries (for variants + the midterm/final)

All ten items are tagged course=MATH120 · week=7 · objective=6 · topic=quadratic-equations and deposited in Item Bank: Week 7 — Quadratic Equations. (Tags: q1 factor-zero-product, q2 zero-product, q3 sqrt-property, q4 sqrt-property-shifted, q5 factor-solve, q6 quadratic-formula, q7 discriminant-negative, q8 discriminant-matching, q9 completing-square, q10 application.)

Canvas placement block

canvas_object   = Quizzes::Quiz
title           = "Week 7 Quiz — Quadratic Equations"
assignment_group = "Quizzes"
points_possible = 10
grading_type    = points
due_offset_days = 6        # 6 days after module start (Sun Oct 18)
published       = true
shuffle_answers = true
provenance      = "~ Prof. Calloway's edition · Fall 2026 · built with thecoursemaker.com"