Quadratic Formula
Deriving and applying x = (−b ± √(b²−4ac)) / 2a. Cash flow, break-even points, and unit economics are common real-world applications of quadratic equations.
Quadratic Formula
The Quadratic Formula
What you'll learn
- Derive and use x = (−b ± √(b² − 4ac)) / 2a.
- Solve when factorisation is difficult.
- Simplify surds in answers where needed.
Key concepts
- Formula — for ax² + bx + c = 0: x = (−b ± √(b²−4ac)) / 2a.
- Identify — a, b, c from standard form (watch signs).
- Substitute — carefully under the square root.
- Simplify — reduce fractions and surds.
- NCERT — solves x² − 4x − 5 = 0 → x = 5 or x = −1.
Worked example
Solve 2x² − 4x − 6 = 0 using the formula
a=2, b=−4, c=−6
x = (4 ± √(16+48)) / 4 = (4 ± 8) / 4
x = 3 or x = −1
Common mistakes
- Sign error on b (if equation is 2x² + 4x − 6, b = +4).
- Forgetting to divide by 2a, not 2.
- Dropping ± and getting only one root.
Quick check
- Write the quadratic formula from memory.
- Solve x² + 6x + 5 = 0 using the formula.
- For x² − 2x + 1 = 0, what happens under the square root?
Open the Practice tab for graded questions on The Quadratic Formula.
Key Takeaways (TL;DR)
- What you'll learn
- Key concepts
- Worked example
- Common mistakes
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