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Class 10 Maths Notes — Chapter 4: Quadratic Equations

Equations of the form ax² + bx + c = 0 — solving techniques and nature of roots.

Detailed NCERT notes

  • Standard form: ax² + bx + c = 0, a ≠ 0.
  • A real number α is a root iff aα² + bα + c = 0. Roots = zeros of the corresponding polynomial.
  • Methods to solve: (1) Factorisation (splitting the middle term), (2) Completing the square, (3) Quadratic formula.
  • Splitting the middle term: find two numbers whose product = a·c and sum = b.
  • Completing the square: rewrite ax² + bx + c = 0 as a(x + b/2a)² = (b² − 4ac)/4a, then take square roots.
  • Quadratic formula: x = [−b ± √(b² − 4ac)] / 2a. Derived from completing the square.
  • Discriminant D = b² − 4ac decides nature of roots:
  • D > 0 → two distinct real roots.
  • D = 0 → two equal real roots (repeated), x = −b/2a.
  • D < 0 → no real roots (complex, out of Class 10 scope).
  • Sum of roots = −b/a; product of roots = c/a (Vieta's).
  • Word problems modelled by quadratics: area/dimension problems, speed/time (with time difference), age (product of ages), streams/pipes, projectile-type numerical.
  • For quadratic word problems, ALWAYS reject the non-physical root (negative length, negative age, negative time).
  • For 'find k for equal roots', set D = 0 and solve for k.
  • For 'real roots exist', require D ≥ 0.

Formulas & key results

  • x = [−b ± √(b² − 4ac)] / 2a
  • D = b² − 4ac
  • Sum = −b/a, Product = c/a

Mind map

  • Factorisation → completing the square → quadratic formula
  • Nature of roots via D
  • Word problems: area, speed-time, age

Tricks & shortcuts

  • Compute D first to know root nature.
  • For 'equal roots', D = 0.
  • In word problems, drop the negative root physically.

Common mistakes to avoid

  • Missing ± in the quadratic formula.
  • Sign errors moving terms across equality.
  • Accepting negative side length or time as an answer.

Competency-based questions & answers

  1. Q. Find k so that kx² − 2kx + 6 = 0 has equal roots.
    A. D = 0: 4k² − 24k = 0 ⇒ k(k − 6) = 0 ⇒ k = 6 (k ≠ 0).
  2. Q. Sum of a number and its reciprocal is 10/3. Find the number.
    A. x + 1/x = 10/3 ⇒ 3x² − 10x + 3 = 0 ⇒ x = 3 or 1/3.
  3. Q. A train covers 360 km at uniform speed. If speed were 5 km/h more, it would take 1 h less. Find speed.
    A. 360/x − 360/(x+5) = 1 ⇒ x² + 5x − 1800 = 0 ⇒ x = 40 km/h (reject −45).