Chapter 4 — Quadratic Equations

Chapter 4 of the Class 10 Maths NCERT textbook, "Quadratic Equations", covers equations of the form ax² + bx + c = 0 (a ≠ 0) and teaches three approaches to find their roots — factorisation, completing the square leading to the quadratic formula, and the discriminant for judging the nature of roots.

• Standard form and roots — A quadratic equation is written as ax² + bx + c = 0 with a not zero, and it can have at most two roots. These roots are exactly the zeroes of the matching quadratic polynomial.
• Methods of solving — Factorisation splits the middle term into two linear factors set to zero, while completing the square leads to the general quadratic formula that works for any solvable equation.
• The discriminant's verdict — The expression b² − 4ac reveals the character of the roots before you compute them: positive gives two distinct real roots, zero gives one repeated root, and negative means no real roots exist.
• Real-world modelling — Many practical situations — hall dimensions, train speeds, products of consecutive integers — reduce to quadratic equations, making these solving techniques broadly useful.

1. 01A quadratic equation in standard form is ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0.
2. 02A quadratic equation has at most two roots; its roots are the same as the zeroes of the quadratic polynomial ax² + bx + c.
3. 03Roots can be found by factorisation: split the middle term, express as a product of two linear factors, and set each factor to zero.
4. 04The quadratic formula gives roots as x = (−b ± √(b²−4ac)) / 2a, valid when b²−4ac ≥ 0.
5. 05The discriminant (b²−4ac) determines the nature of roots: >0 means two distinct real roots, =0 means two equal real roots, <0 means no real roots.
6. 06Quadratic equations model many real-life situations such as dimensions of halls, speed of trains, and product of consecutive integers.