A complex number is z = a + bi, where a is the real part, b is the imaginary part, and i² = −1.
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Complex Numbers and Quadratic Equations
Concept capsules, formulas, guided nudges, speed drills, a coach walkthrough, and a mini test for this chapter.
Core ideas
Modulus |z| = √(a²+b²); it measures the distance of z from the origin in the Argand plane.
Conjugate of z = a+bi is z̄ = a−bi; z·z̄ = a²+b² = |z|².
The quadratic formula x = [−b ± √(b²−4ac)]/2a gives both roots of ax²+bx+c=0.
If the discriminant b²−4ac < 0, the roots are complex conjugates.
Argand Plane Explorer
Reason it out
1. Find the value of i²⁰.
Need a nudge?
Powers of i cycle every 4: i,−1,−i,1.
Explanation: i²⁰=(i⁴)⁵=1⁵=1.
2. Find |3+4i|.
Need a nudge?
Use |z|=√(a²+b²).
Explanation: |3+4i|=√(9+16)=√25=5.
3. For x²+2x+5=0, what does the discriminant tell you about the roots?
Need a nudge?
Discriminant = b²−4ac.
Explanation: b²−4ac = 4−20 = −16 < 0, so the roots are complex conjugates.
4. For x²−7x+12=0, find the sum and product of roots.
Need a nudge?
Sum=−b/a.
Explanation: Sum=−(−7)/1=7; product=12/1=12.
5. Find the modulus and argument of z = 1 + i.
Need a nudge?
Compute r = √(x²+y²) with x=1, y=1.
Explanation: r=√(1²+1²)=√2. Since cosθ=1/√2 and sinθ=1/√2, θ=π/4 (both x,y positive, so first quadrant).
Fast recognition
Find i¹⁰.
Target: 15 seconds
Quick method
i¹⁰=i^(8+2)=i²=−1.
Find |6+8i|.
Target: 15 seconds
Quick method
6-8-10 triple: √(36+64)=10.
Find i^25.
Target: 10 seconds
Quick method
25 mod 4=1, so i^25=i.
The modulus of z = 3 − 4i is:
Target: 20 seconds
Quick method
|z|=√(a²+b²) regardless of sign of b.
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