Quantum Complex Answer Key
Quantum Key
Answer Key: Solving Quadratic Equations (Complex)
Teacher Guide
1. x² + 4x + 13 = 0 (Quadratic Formula)
x = -2 ± 3i
Step 1: Identify & Substitute
a = 1, b = 4, c = 13
x = [-4 ± √(4² - 4(1)(13))] / 2(1)
Step 2: Simplify Discriminant
x = [-4 ± √(16 - 52)] / 2
x = [-4 ± √(-36)] / 2 = (-4 ± 6i) / 2
x = -2 ± 3i
2. x² - 6x + 25 = 0 (Completing Square)
x = 3 ± 4i
Step 1: Complete the Square
x² - 6x = -25
x² - 6x + 9 = -25 + 9
(x - 3)² = -16
Step 2: Take Root & Solve
x - 3 = ±√(-16)
x - 3 = ±4i
x = 3 ± 4i
3. 2x² + 12x + 26 = 0 (Quadratic Formula)
x = -3 ± 2i
Step 1: Optional Simplify
Divide by 2: x² + 6x + 13 = 0
Step 2: Formula Application
x = [-6 ± √(36 - 52)] / 2
Step 3: Simplify Results
x = [-6 ± √(-16)] / 2
x = [-6 ± 4i] / 2
x = -3 ± 2i
4. x² + 10x + 34 = 0 (Completing Square)
x = -5 ± 3i
Step 1: Isolate & Square
x² + 10x + 25 = -34 + 25
(x + 5)² = -9
Step 2: Complex Root
x + 5 = ±3i
x = -5 ± 3i
5. x² - 2x + 10 = 0 (Any Method)
x = 1 ± 3i
Method: Completing Square
x² - 2x + 1 = -9
(x - 1)² = -9
Final Solution
x - 1 = ±3i
x = 1 ± 3i
Common Pitfalls to Watch For
01
Negative Discriminant: Students often forget the i when taking the square root of a negative number.
02
Completing the Square: Remind students to add (b/2)² to BOTH sides of the equation.
03
Simplify First: In Problem 3, emphasize that dividing by 2 first makes the Quadratic Formula or Completing the Square much easier.
04
Signs: Watch for sign errors when moving terms (e.g., subtracting c from both sides).
