Class 10 Maths Chapter 4: Quadratic Equations– NCERT Notes & Formula

Class 10 NCERT Maths Chapter 4: Quadratic Equations

1. Introduction

In earlier classes, you studied polynomials and their degrees. In this chapter, we focus on quadratic equations — equations of the form:

\[ ax^2 + bx + c = 0,\quad a \ne 0 \]

We learn methods to find their roots and understand their applications in real-life problems.

2. Topics in the 2025–26 Syllabus

  • Standard form of a quadratic equation
  • Solutions of a quadratic equation by:
    • Factorisation
    • Completing the square (derivation of formula)
    • Quadratic formula
  • Nature of roots using the discriminant \(D = b^2 - 4ac\)
  • Simple real-life application problems

Note: Problems with equations reducible to quadratic form are removed from the latest syllabus.

3. Methods of Solving Quadratic Equations

Factorisation Method:

Example: \[ x^2 - 5x + 6 = 0 \] Factorise: \((x - 2)(x - 3) = 0\) So, \(x = 2\) or \(x = 3\).

Completing the Square:

Example: \[ x^2 + 6x - 16 = 0 \] Move constant: \(x^2 + 6x = 16\) Add \(9\) on both sides: \(x^2 + 6x + 9 = 25\) \[ (x + 3)^2 = 25 \quad\Rightarrow\quad x + 3 = \pm 5 \] So, \(x = 2\) or \(x = -8\).

Quadratic Formula:

For \(ax^2 + bx + c = 0\), the roots are: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

4. Nature of Roots

The discriminant \(D = b^2 - 4ac\) decides the nature of roots:

\[ \begin{cases} D > 0 &\Rightarrow \text{Two distinct real roots} \\ D = 0 &\Rightarrow \text{Two equal real roots} \\ D < 0 &\Rightarrow \text{No real roots} \end{cases} \]

5. Example Word Problem

The product of two consecutive positive integers is 306. Find the integers.

Let the integers be \(n\) and \(n+1\): \[ n(n+1) = 306 \quad\Rightarrow\quad n^2 + n - 306 = 0 \] Solve by factorisation: \((n - 17)(n + 18) = 0\) So, \(n = 17\) (positive integer), hence the integers are 17 and 18.

6. Summary

  • Quadratic equations have degree 2 and are written in standard form.
  • Three solving methods: factorisation, completing the square, quadratic formula.
  • Nature of roots depends on the discriminant.
  • Application problems connect algebra to real-world situations.
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