Quadratic Equations (Class 10)
About 6 min · Mathematical Reasoning · English
Time worked out from the lesson's 419 words at 150 words a minute, plus a minute for each question.
Quadratic equations form a critical bridge between linear algebra and higher mathematics in the SOF IMO curriculum. A quadratic equation is any equation of the form ax² + bx + c = 0, where a ≠ 0.
Key ideas
- Standard form: A quadratic equation must be written as ax² + bx + c = 0 (a ≠ 0) before solving. If a = 0, it becomes linear, not quadratic.
- Roots or zeroes: The values of x that satisfy the equation. A quadratic can have at most two real roots, though they may be equal or complex.
- Factorization method: Express ax² + bx + c as a product of two linear factors. Works cleanly only when roots are rational. Look for two numbers that multiply to ac and add to b.
- Completing the square: Transform the equation into (x + p)² = q form. This method reveals the vertex of the parabola and works for all quadratics, though it's algebraically intensive.
- Quadratic formula: x = (–b ± √(b² – 4ac)) / 2a. Universal method that works for any quadratic. Memorize this formula—it's your fallback for difficult factorizations.
Watch out for
- Forgetting a ≠ 0: Writing 0x² + 3x + 5 = 0 is linear, not quadratic. Always check the coefficient of x² before applying quadratic methods.
- Sign errors in the quadratic formula: The formula is –b, not +b. Many students write x = (b ± √Δ)/2a and lose both roots. Write the negative sign explicitly.
- Incorrect discriminant interpretation: Δ = 0 means two equal roots, not one root. The root is repeated. Also, Δ < 0 means no real solution, but some students attempt to simplify √(–4) within real numbers.
Quick reference
- Quadratic form: ax² + bx + c = 0; ensure a ≠ 0
- Factorization: Works when roots are rational; find factors of ac that add to b
- Formula: x = (–b ± √(b² – 4ac)) / 2a — universal method
- Discriminant: Δ = b² – 4ac tells root nature before solving
- Sum/Product of roots: –b/a and c/a respectively — use to verify or form equations
Check yourself: 3 questions
1. If one root of the quadratic equation x² - 7x + k = 0 is 3, then what is the value of k?
2. For what value of m will the equation (m - 2)x² + 8x + m + 4 = 0 have equal roots?
3. If the roots of the equation x² - 7x + k = 0 are equal, what is the value of k?
This lesson is cut from the topic's note, not rewritten. Fact-checked on 21 Oct 2026: an AI-assisted check of this note against cited sources; the errors it found were corrected that day. Some errors may remain. The questions are from Shishya's validated question bank.
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