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## 1. Equation: ax^2 + bx +c = 0

### 1.1. Concepts

1.1.1. Use α^2 +β^2, αβ as algebraic identities

1.1.2. Do not solve for α&β

### 1.2. Formats

1.2.1. x^2 - (α+β)x +αβ = 0

1.2.2. x^2 -(sum of alpha + beta)x+(product of alpha and beta)= 0

### 2.1. Keywords

2.1.1. Real & distinct

2.1.1.1. b^2 -4ac > 0

2.1.2. Equal

2.1.2.1. b^2 -4ac = 0

2.1.3. curve above/below the x-axis ( always)

2.1.3.1. b^2 -4ac < 0

2.1.4. tangent

2.1.4.1. b^2 -4ac = 0

2.1.5. no real roots

2.1.5.1. b^2 - 4ac< 0

2.1.6. meets

2.1.6.1. b^2 -4ac ≥ 0

2.1.7. Real roots

2.1.7.1. b^2 - 4ac≥0

### 2.2. Basic curve

2.2.1. Solve b^2-4ac = 0

2.2.2. Solve b^2-4ac < 0

2.2.3. Solve b^2-4ac > 0

### 2.3. Curve and line

2.3.1. Simultaneous equations

2.3.2. ax^2 + bx + c = 0

2.3.3. Solve b^2-4ac = 0

2.3.4. Solve b^2-4ac < 0

2.3.5. Solve b^2-4ac > 0

2.3.6. Solve

### 2.4. PROVE IT

2.4.1. cannot set b^2 -4ac > 0

2.4.1.1. Use algebra to prove

2.4.2. cannot set b^2 -4ac = 0

2.4.2.1. Use algebra to prove

2.4.3. cannot set b^2 -4ac < 0

2.4.3.1. Use algebra to prove