📈 Lesson 5: Quadratic Equations

Projectile motion, energy, and anywhere acceleration is involved

🎯 Learning Objectives

• Solve quadratics by factoring, quadratic formula, and completing the square
• Interpret the discriminant (b² - 4ac)
• Apply quadratics to physics: projectile time-of-flight, max height, energy problems
• Reject non-physical solutions (negative time, imaginary distance)

🔑 Key Concepts

Standard Form

ax² + bx + c = 0 where a ≠ 0

Three Solution Methods

Method	When to Use
Factoring	Simple integer coefficients, obvious factors
Quadratic Formula	Always works: x = (-b ± √(b²-4ac))/(2a)
Completing Square	Deriving formulas, vertex form

The Discriminant: Δ = b² - 4ac

• Δ > 0: Two distinct real solutions
• Δ = 0: One real solution (double root)
• Δ < 0: No real solutions (complex) — in physics, this often means "impossible scenario"

Physics Context

Quadratics appear when:

• Position has t² term: x = x₀ + v₀t + ½at²
• Energy has v² term: KE = ½mv²
• Projectile motion: y = y₀ + v₀sinθ·t - ½gt²

📐 Worked Examples

Example 1: Factoring — Time When Object Returns to Ground

y = 20t - 5t². When does it hit the ground (y=0)?

0 = 20t - 5t²
0 = 5t(4 - t)      ← Factor out 5t
t = 0 or t = 4 s   ← Zero product property
            

t=0 is launch; t=4s is landing.

Example 2: Quadratic Formula — Projectile Height

y = 2 + 15t - 4.9t². When does it reach 10m height?

10 = 2 + 15t - 4.9t²
4.9t² - 15t + 8 = 0    ← Standard form
a=4.9, b=-15, c=8
t = (15 ± √(225 - 156.8))/9.8
t = (15 ± √68.2)/9.8
t = (15 ± 8.26)/9.8
t₁ = 2.37 s (going up)
t₂ = 0.69 s (going down)
            

Example 3: Energy — Speed from Height

mgh = ½mv² + mgh'. Mass cancels. v = √(2g(h-h')). Find h' when v=10, g=10, h=20.

10 = √(2·10·(20-h'))
100 = 20(20-h')
5 = 20 - h'
h' = 15 m
            

🏋️ Practice Problems

1. Solve: t² - 6t + 8 = 0 (factor)
2. Solve: 4.9t² - 20t + 15 = 0 (quadratic formula)
3. Solve: v² - 100 = 0 (difference of squares)
4. Ball thrown up: y = 1.5 + 20t - 4.9t². When at ground level?
5. Same ball: When at y = 15m? (two answers — explain)
6. Range equation: R = v₀²sin(2θ)/g. Find θ when R=20, v₀=20, g=10 (solve sin(2θ) = 0.5)
7. KE = ½mv². If KE doubles, by what factor does v increase? (quadratic relationship)
8. Free fall from 80m: 4.9t² = 80. Find t.

✅ Answers

Click to reveal answers

1) t=2, 4 2) t=1.07s, 2.85s 3) v=±10 4) t≈4.29s 5) t≈0.83s and 3.67s (up/down)

6) 2θ=30° or 150° → θ=15° or 75° 7) v increases by √2 ≈ 1.414 8) t≈4.04s

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