NCERT Class 10 Maths – Quadratic Equations Concepts with Examples

Understanding Quadratic Equations is a crucial part of Class 10 Mathematics. Whether you’re aiming to score high in the CBSE Board Exams or just trying to strengthen your foundation, mastering this chapter is a must. In this complete guide, we’ll break down all the key concepts of Quadratic Equations in a simple and clear way — including standard forms, methods of solving, real-life applications, and important formulas.

This guide is based on the latest CBSE syllabus and NCERT guidelines and is perfect for quick revision, doubt-clearing, or preparing for school tests and board exams.

NCERT Class 10 Maths – Quadratic Equations Concepts with Examples

Quadratic Equations form a key part of the Class 10 NCERT Maths syllabus and are essential for building a strong mathematical foundation. This chapter introduces students to equations of the second degree and explores various methods to solve them with ease.

In this guide, we’ll cover all the core concepts of Quadratic Equations, including definitions, standard form, methods of solving, nature of roots, and real-life word problems — all explained with clear examples. Whether you’re preparing for your CBSE board exams or revising for school tests, this complete breakdown will help you learn quickly and confidently.

Let’s begin by understanding what exactly a quadratic equation is.

Quadratic Equations Concepts for Class 10 – Complete Guide for CBSE Students

Quadratic Equations is one of the most important chapters in Class 10 Maths. It not only builds upon your earlier understanding of algebra but also forms the foundation for many advanced topics in mathematics. Whether you’re preparing for your board exams or building a solid base for competitive exams like JEE, NTSE, or Olympiads, mastering the concepts of quadratic equations is essential.

In this blog post, we’ll cover all important concepts from NCERT Chapter 4 – Quadratic Equations in a student-friendly way.

📘 What is a Quadratic Equation?

A quadratic equation is a second-degree polynomial equation in one variable. It is of the form

ax² + bx + c = 0

Where:

• a, b, and c are real numbers, and
• a ≠ 0 (because if a = 0, it becomes a linear equation).

The variable x represents an unknown quantity, and the term x² shows that it’s a second-degree equation.

🎯 Standard Form

The standard form of a quadratic equation is:

ax² + bx + c = 0

Example:

• 2x² + 5x - 3 = 0 → Already in standard form.
• x(x + 4) = 5 → Convert to standard form: x² + 4x - 5 = 0

🧠 Roots of a Quadratic Equation

The roots (or solutions) of a quadratic equation are the values of x which satisfy the equation ax² + bx + c = 0.

Roots are the points where the graph of the quadratic equation cuts the x-axis.

📌 Methods of Solving Quadratic Equations

There are three primary methods to solve a quadratic equation:

1. Factorisation Method

This is the simplest method and is applicable when the quadratic can be factored easily.

Example:

x² - 7x + 12 = 0
=> (x - 3)(x - 4) = 0
=> x = 3, x = 4
2. Completing the Square Method
This method involves converting the quadratic expression into a perfect square.

Example:x² + 6x + 5 = 0
=> x² + 6x = -5
=> x² + 6x + 9 = 4 (add 9 both sides)
=> (x + 3)² = 4
=> x + 3 = ±2
=> x = -1, -5
3. Quadratic Formula Method
This is a universal method and works for any quadratic equation. The formula is:
x = (-b ± √(b² - 4ac)) / 2a
Where a, b, and c are the coefficients from the equation ax² + bx + c = 0.
Example:2x² - 4x + 1 = 0
a = 2, b = -4, c = 1
D = b² - 4ac = 16 - 8 = 8
x = (4 ± √8) / 4 = (4 ± 2√2) / 4 = (2 ± √2)/2
Discriminant and Nature of Roots
The discriminant (D) helps determine the nature of roots of the quadratic equation.
D = b² - 4ac
Discriminant D	Nature of Roots
D > 0	Two distinct real roots
D = 0	Two equal real roots
D < 0	Imaginary roots

📉 Graphical Representation
The graph of a quadratic equation is a parabola.

If a > 0, the parabola opens upward.

If a < 0, the parabola opens downward.

The roots of the equation are the x-intercepts of the parabola (where the graph cuts the x-axis).

💡 Real-Life Applications of Quadratic Equations
Quadratic equations are not just theoretical—they are used in many practical situations:

Projectile motion in physics

Maximizing or minimizing profit in business

Area optimization problems in architecture

Finding speed and distance

Solving geometry word problems

📘 NCERT-Based Exercises for Chapter 4

The NCERT textbook divides this chapter into the following exercises:

Exercise 4.1 – Introduction and identifying quadratic equations

Exercise 4.2 – Solving by Factorisation

Exercise 4.3 – Solving using the Quadratic Formula

Exercise 4.4 – Nature of Roots

Optional Exercise – Word problems and applications

👉 You can find step-by-step solutions for all these exercises on our website:
🔗 mathscience.in

✨ Quick Tips for Students
Memorize the standard form and quadratic formula

Understand the meaning of discriminant

Practice all three solving methods

Focus on word problems from NCERT

Attempt previous year board questions

📚 Summary
Let’s quickly revise what we learned:

A quadratic equation is in the form ax² + bx + c = 0.

It can be solved using factorisation, completing the square, or quadratic formula.

The discriminant tells us the type of roots.

Quadratic equations have real-life applications in science, business, and engineering.

Mastery of this topic is essential for success in board exams and beyond.

📢 Final Words
Understanding Quadratic Equations is not just about solving problems—it's about thinking logically and applying math in the real world. Whether you're aiming for top marks or preparing for entrance exams, this topic will follow you through higher classes too.

Stay consistent with your practice and don’t forget to visit:

🔗 mathscience.in – Your learning partner for Class 9 and 10 Science and Maths!

🔗 Official NCERT Textbook PDF for Class 10 Maths Chapter 4 – Quadratic Equations:
👉 https://ncert.nic.in/textbook.php?jemh1=4-10
(Credit: NCERT Official Website – ncert.nic.in)