The Quadratic Formula
Quadratic Formula — ax² + bx + c = 0x = [−b ± √(b² − 4ac)] ÷ 2a
Discriminant Δ = b² − 4ac
Discriminant Δ = b² − 4ac
Where a, b, c are real number coefficients and a ≠ 0. The ± gives two roots: one using +√ and one using −√.
Types of Roots Based on Discriminant
| Discriminant (Δ) | Type of Roots | Example |
|---|---|---|
| Δ > 0 | Two distinct real roots | x²−5x+6=0 → x=3, x=2 |
| Δ = 0 | One real root (repeated) | x²−2x+1=0 → x=1 |
| Δ < 0 | Two complex conjugate roots | x²+x+1=0 → complex |
Step-by-Step Example
Solve: 2x² − 7x + 3 = 0
a=2, b=−7, c=3
Δ = (−7)² − 4×2×3 = 49 − 24 = 25
√Δ = 5
x₁ = (7 + 5) ÷ 4 = 3
x₂ = (7 − 5) ÷ 4 = 0.5
Verification: 2(3)²−7(3)+3 = 18−21+3 = 0 ✓
💡 The sum of roots = −b/a and product of roots = c/a. These properties are useful for quick verification without substituting back.
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