A fast, one-glance recap of Class 10 Maths Chapter 4 (Quadratic Equations) — for the full worked explanations, see the Solutions post.
Revision Notes: Quadratic Equations
- Standard form: ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0
- Quadratic equation: a polynomial equation of degree 2 in one variable — always check the highest power of x survives after simplification before calling something “quadratic”
- Root/solution: a real number α is a root of ax²+bx+c=0 if aα²+bα+c = 0; a quadratic equation has at most two roots
- Method 1 — Factorisation: split the middle term bx into two terms whose coefficients multiply to give ac and add to give b, then factor by grouping; each factor set to zero gives a root
- Method 2 — Quadratic formula: x = [−b ± √(b²−4ac)] / 2a
- Discriminant: D = b² − 4ac — determines the nature of the roots without fully solving the equation
- D > 0 → two distinct real roots
- D = 0 → two equal real roots (repeated root), x = −b/2a
- D < 0 → no real roots
- Condition for equal roots: b² = 4ac (used to solve for an unknown parameter k, p, m, etc. in a given equation)
- Sum and product of roots (if roots are α, β): sum α+β = −b/a, product αβ = c/a — useful for reverse-engineering an equation from its roots: x² − (sum)x + (product) = 0
- Irrational/surd roots of a quadratic with rational coefficients always occur in conjugate pairs, e.g. if one root is p+√q, the other is p−√q
- Word problem method: define the unknown clearly (age, speed, number, side length, etc.), translate every condition in the problem into an algebraic expression, form a single quadratic equation, solve it, and always reject roots that don’t fit the real-world context (negative age, negative length, negative speed, etc.)
- Common problem patterns: consecutive integers, two-part sum/product problems, age problems (present age vs. age n years ago/hence), speed–distance–time (uniform speed changed by some amount), area/perimeter of rectangles, right-triangle side relationships (Pythagoras), and cost-of-production problems
- Checking a solution: always verify by substituting the found value(s) back into the original word conditions, not just the equation, to catch sign/setup errors
See also: Extra Questions (HOTS) | Formulas Handbook
