The Quadratic Formula
2. Identify a, b, c (watch negative signs!).
3. Calculate Discriminant: Δ = b² - 4ac.
4. Split into ± branches and simplify.
| Case / Type | Step-by-Step Calculation | Final Exact Roots |
|---|---|---|
| 1. Rational Roots 2x² + 3x - 2 = 0 | a=2, b=3, c=-2 ➔ Δ = 3² - 4(2)(-2) = 9 + 16 = 25 x = (-3 ± √25) / 4 = (-3 ± 5) / 4 x₁ = (-3 + 5)/4 = 2/4 = 0.5 | x₂ = (-3 - 5)/4 = -8/4 = -2 | x = 0.5 x = -2 |
| 2. Radical Roots x² - 4x + 1 = 0 | a=1, b=-4, c=1 ➔ Δ = (-4)² - 4(1)(1) = 16 - 4 = 12 x = (-(-4) ± √12) / 2 = (4 ± 2√3) / 2 Divide both terms by 2 ➔ x = 2 ± √3 | x = 2 ± √3 ≈ 3.732, 0.268 |
| 3. Complex Roots x² - 2x + 5 = 0 | a=1, b=-2, c=5 ➔ Δ = (-2)² - 4(1)(5) = 4 - 20 = -16 √(-16) = √(16) · i = 4i x = (-(-2) ± 4i) / 2 = (2 ± 4i) / 2 = 1 ± 2i | x = 1 ± 2i |
b = -5, -b = +5, NOT -5!• Trap 2 (Fraction Bar): Divide ALL terms by
2a, not just the square root.• Trap 3 (Squaring Negative):
(-4)² = +16. A squared real number is ALWAYS positive!• Δ = 0: 1 Repeated Real Root (Vertex touches x-axis at 1 point).
• Δ < 0: 2 Complex Conjugate Roots (Parabola floats above/below x-axis).
The Quadratic Formula: Step-by-Step Guide, Visual Solver & Exam Traps
The Swiss Army Knife of Algebra (When Factoring Fails)
Factoring only works when equations have neat, friendly whole numbers like (x - 2)(x - 3) = 0. But in real-world physics, finance, and engineering, numbers are messy decimals and non-whole square roots (like √17).
The Quadratic Formula is the universal multi-tool: feed it ANY three numbers a, b, c and it is mathematically guaranteed to output the exact solutions 100% of the time!
How to Execute the Formula (With a Running Example)
Let's follow each step using the equation x² - 5x + 6 = 0:
ax² + bx + c = 0.• Our Example: In
1x² - 5x + 6 = 0, we get: a = 1, b = -5, c = 6.• Our Example:
Δ = (-5)² - 4(1)(6) = 25 - 24 = 1.• Our Example:
√Δ = √1 = 1.x₁ = (-(-5) + 1) / (2 · 1) = (5 + 1) / 2 = 6 / 2 = 3• Minus Road (-):
x₂ = (-(-5) - 1) / (2 · 1) = (5 - 1) / 2 = 4 / 2 = 2The 3 Mistakes That Cost Students Marks on Exams
When b = -6, the formula gives -(-6) = +6. Students frequently write -6 by mistake. Always wrap negative numbers in parentheses!
The fraction line extends under ALL of -b ± √Δ. Dividing only the square root by 2a is an instant algebra failure.
(-5)² = +25, NOT -25. The square of any real number inside b² is ALWAYS positive.
3 Step-by-Step Worked Examples
Master the exact derivation across rational, radical, and complex solutions:
🎯 Active Recall Quiz (Test Your Formula Mastery)
Test your algebraic intuition with 3 quick questions. Instant feedback provided:
Frequently Asked Questions
The quadratic formula x = (-b ± √(b² - 4ac)) / (2a) is the universal method to find the exact roots (x-intercepts) of ANY quadratic equation in standard form ax² + bx + c = 0, even when factoring is impossible.
If b is negative (e.g., b = -5), then -b becomes positive +5 because -(-5) = +5. This is the #1 most common student error on algebra exams.
The ± symbol indicates two distinct branches: one where you ADD the square root (x₁ = (-b + √Δ) / 2a) and one where you SUBTRACT the square root (x₂ = (-b - √Δ) / 2a).
The entire numerator (-b ± √(b² - 4ac)) must be divided by 2a. Dividing only the radical term by 2a and forgetting to divide -b is a common algebraic error.
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