Schooling · Class 10 · Mathematics · Quadratic Equations · Chapter on a page
CBSE Class 10 · Mathematics · NCERT Mathematics
Quadratic Equations: chapter on a page
A cut of Shishya's notes for this chapter: the first paragraph of the introduction and the key ideas, formulas and facts, common mistakes and quick revision lists, word for word. Nothing is added or reworded; the worked examples are left out (they are on the chapter page).
About the notes: Written by Shishya's AI on 26 Sept 2026 from the chapter's title and class level, in Shishya's own words — not a copy or summary of the textbook. Read the official chapter for the book's own text, activities and exercises.
This note's current text has no applied fact-check on record.
Mind map
Drawn from the same notes by a fixed rule (no AI): each label is the item's own words — its bold term, its label before a colon, a short formula as written, or its first words.
The mind map as a list
- Key ideas
- A quadratic equation in variable x…
- The values of x that satisfy the…
- Factorisation method works when ax²…
- Completing the square converts any…
- The quadratic formula gives roots…
- The discriminant D = b² − 4ac…
- A quadratic equation can have at…
- Formulas and facts
- Standard form
- Quadratic formula
- Discriminant
- Nature of roots based on D
- D > 0
- D = 0
- D < 0
- Watch out for
- Writing ax² + bx + c = 0 with a = 0…
- Forgetting the ± sign in the…
- Taking square root of a negative…
- Accepting negative values for…
- Errors in sign when calculating b² −…
What this chapter is about
A quadratic equation is a polynomial equation of degree two. After learning linear equations in earlier classes, you now move to equations where the variable appears with a highest power of two. These equations arise naturally when dealing with areas, projectile motion, profit-loss calculations and many situations where the relationship between quantities is not simply proportional.
Key ideas
- A quadratic equation in variable x has the standard form ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0.
- The values of x that satisfy the equation are called roots or solutions; a quadratic equation has at most two roots.
- Factorisation method works when ax² + bx + c can be written as a product of two linear factors.
- Completing the square converts any quadratic into the form (x + p)² = q, from which roots can be found by taking square roots.
- The quadratic formula gives roots directly: x = (−b ± √(b² − 4ac)) / 2a, and works for every quadratic equation.
- The discriminant D = b² − 4ac determines the nature of roots: if D > 0, two distinct real roots; if D = 0, two equal real roots; if D < 0, no real roots.
- A quadratic equation can have at most two real roots, never three or more.
Formulas and facts to remember
- Standard form: ax² + bx + c = 0, where a ≠ 0.Meaning: The equation must have a non-zero coefficient for x².
- Quadratic formula: x = (−b ± √(b² − 4ac)) / 2aMeaning: Substitute values of a, b, c to find both roots directly.
- Discriminant: D = b² − 4acMeaning: This single number tells you about the nature of roots.
- Nature of roots based on D:
- D > 0 → two distinct real roots
- D = 0 → two equal real roots (also called repeated root)
- D < 0 → no real roots
- Sum of roots: −b/aProduct of roots: c/aMeaning: These relate coefficients to roots without solving.
Common mistakes
- Writing ax² + bx + c = 0 with a = 0 and calling it quadratic → Remember, a must be non-zero; otherwise it becomes linear.
- Forgetting the ± sign in the quadratic formula and finding only one root → Always compute both (−b + √D)/2a and (−b − √D)/2a.
- Taking square root of a negative discriminant and writing a real answer → If D < 0, state that no real roots exist; do not proceed further.
- Accepting negative values for length, age or count in word problems → Check whether the root makes sense in the given context and reject impossible values.
- Errors in sign when calculating b² − 4ac, especially when b or c is negative → Write out each term carefully before combining.
Quick revision
- Standard form: ax² + bx + c = 0 with a ≠ 0.
- Quadratic formula: x = (−b ± √(b² − 4ac)) / 2a.
- D > 0 means two different real roots; D = 0 means one repeated root; D < 0 means no real roots.
- Always check if roots are valid for the given word problem context.
- Factorisation is quick when factors are easy to spot; otherwise use the formula.