Quadratic Equations Class 10 Notes (2026-27) — CBSE
Class 10 Maths Chapter 4 notes: standard form, solving by factorisation and the quadratic formula, the discriminant and nature of roots, with word problems.
Quadratic Equations — Class 10 Maths Notes
Chapter Snapshot
A quadratic equation is a polynomial equation of degree 2. This chapter teaches you to recognise the standard form, solve it by factorisation and by the quadratic formula, use the discriminant to decide the nature of its roots, and form quadratic equations from word problems.
Board relevance: a scoring Algebra chapter. Expect a solving question, a "nature of roots" or "find k for equal roots" question, and a word problem (often on ages, speed, or areas). Together these are worth about 5–6 marks.
Key Concepts & Definitions
Standard form: ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0.
- a = coefficient of x², b = coefficient of x, c = constant term.
- If a = 0 the equation is not quadratic (it becomes linear).
Root (or zero) of a quadratic equation — a value of x that satisfies it. A quadratic equation has at most two roots. If α is a root, then aα² + bα + c = 0.
Formulas
Method 1 — Factorisation (splitting the middle term)
1. Write in standard form.
2. Split the middle term b into two numbers whose product = a × c and sum = b.
3. Factor by grouping into (x − α)(x − β) type factors.
4. Set each factor to zero → the two roots.
Method 2 — Quadratic formula
Derived by completing the square on ax² + bx + c = 0:
x = [ −b ± √(b² − 4ac) ] / 2a
Valid whenever b² − 4ac ≥ 0. Use it when factorisation is difficult.
The discriminant and nature of roots
D = b² − 4ac
Discriminant Nature of roots
D 0 Two distinct real roots
D = 0 Two equal (repeated) real roots, each = −b/2a
D < 0 No real roots (roots are imaginary)
Sum and product of roots
If α and β are the roots of ax² + bx + c = 0:
- Sum: α + β = −b/a
- Product: αβ = c/a
- To form a quadratic with given roots: x² − (α + β)x + αβ = 0.
Worked Examples
Example 1 — Factorisation: Solve 6x² − x − 2 = 0.
Product a×c = 6×(−2) = −12; need two numbers with product −12, sum −1 → −4 and 3.
6x² − 4x + 3x − 2 = 0 → 2x(3x − 2) + 1(3x − 2) = 0 → (3x − 2)(2x + 1) = 0.
Roots: x = 2/3 and x = −1/2.
Example 2 — Quadratic formula: Solve 2x² − 7x + 3 = 0.
a = 2, b = −7, c = 3. D = 49 − 24 = 25 0.
x = [7 ± √25]/4 = [7 ± 5]/4 → x = 3 or x = 1/2.
Example 3 — Nature / find k: For what value of k does kx² − 2kx + 6 = 0 have equal roots?
Equal roots ⇒ D = 0: (−2k)² − 4(k)(6) = 0 → 4k² − 24k = 0 → 4k(k − 6) = 0.
k = 0 is rejected (then it is not quadratic), so k = 6.
Example 4 — Word problem: The product of two consecutive positive integers is 306. Find them.
Let the integers be x and x + 1: x(x + 1) = 306 → x² + x − 306 = 0 → (x + 18)(x − 17) = 0.
x = 17 (reject −18, must be positive). The integers are 17 and 18.
Example 5 — Speed word problem: A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less. Find the speed.
Let the speed be x km/h. Time = distance/speed, so 360/x − 360/(x + 5) = 1.
Multiply through by x(x + 5): 360(x + 5) − 360x = x(x + 5) → 1800 = x² + 5x → x² + 5x − 1800 = 0.
Factorise: (x + 45)(x − 40) = 0 → x = 40 (reject −45). The speed is 40 km/h.
This "time = distance ÷ speed" set-up, leading to a quadratic after clearing denominators, is one of the most common board word problems.
Graphical meaning of the roots: the graph of y = ax² + bx + c is a parabola. Its real roots are the x-values where the parabola cuts the x-axis. If D 0 it cuts the axis at two points, if D = 0 it just touches the axis at one point (equal roots), and if D < 0 it does not meet the axis at all (no real roots). This links the algebra of the discriminant to what you would see on a graph.
Important Question Patterns
1. Solve (2–3 marks): by factorisation or the quadratic formula; always reduce to standard form first.
2. Nature of roots (2 marks): compute D and state the nature; or set D = 0 (equal roots) / D 0 / D < 0 to find an unknown coefficient.
3. Word problems (3–4 marks): ages, consecutive numbers, speed–distance–time (e.g. a train's usual speed), areas/dimensions of a rectangle, work/time. Form the equation, solve, and reject the inadmissible root (e.g. negative length or speed).
4. Check whether an equation is quadratic (1 mark): simplify and see if the x² term survives with a ≠ 0.
5. Form an equation (2 marks): from given roots using x² − (sum)x + product = 0.
⚡ Quick Revision
- Standard form: ax² + bx + c = 0, a ≠ 0. At most two roots.
- Factorisation: split middle term into two numbers with product a·c and sum b.
- Quadratic formula: x = [−b ± √(b² − 4ac)] / 2a.
- Discriminant D = b² − 4ac: D 0 → two distinct real; D = 0 → equal real (−b/2a); D < 0 → no real roots.
- Sum of roots = −b/a, product = c/a; form an equation as x² − (sum)x + product = 0.
- Word problems: define the variable, form the equation, solve, and reject the root that doesn't fit (negative length/speed/age).
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