Chapter 4: Quadratic Equations Notes for Class 10 - The Complete Guide

Quadratic Equations for Class 10 Maths Notes are available in this article.This guide explains quadratic equations in a simple, exam-friendly way. Aligned with the NCERT syllabus and CBSE exam pattern, it covers the standard form and three main methods to solve them: factorization, completing the square, and the quadratic formula. You’ll also learn how the discriminant reveals the nature of roots, and how to approach real-life word problems that use quadratics.

Each topic includes step-by-step worked examples, useful shortcut tips for faster factorization, and quick ways to check your answers. Short revision bullets and practice problems with answers are provided to help you build confidence before tests. You can download a complete PDF version of these Quadratic Equations notes for offline study. 

Chapter 4: Quadratic Equations Notes for Class 10 

1. What Is a Quadratic Equation

A quadratic equation is any equation of the form:

ax² + bx + c = 0

where a, b, and c are real numbers, and a ≠ 0.

Where, 

a = coefficient of x² (also called the quadratic coefficient). It cannot be zero. If it were, the equation would just be linear.

b = coefficient of x (the linear coefficient)

c = the constant term

Converting to Standard Form

Sometimes a quadratic equation won't look like ax² + bx + c = 0. You need to rearrange it. For example:

Example: Is x(x − 4) = −3 a quadratic equation?

Expand: x² − 4x = −3

Rearrange: x² − 4x + 3 = 0 

Yes, it's quadratic in standard form with a = 1, b = −4, c = 3.

2. Roots of a Quadratic Equation

The roots (also called solutions or zeroes) of a quadratic equation ax² + bx + c = 0 are the values of x that make the equation true.

A quadratic equation can have the following:

• Two distinct real roots

• Two equal real roots (also called a repeated root)

• No real roots (the roots are complex/imaginary)

 

Graphical Meaning

When you plot the quadratic polynomial y = ax² + bx + c, you get a parabola.

• 
  If the parabola crosses the x-axis at two points ⇒ two distinct real roots

• If it just touches the x-axis at one point ⇒ two equal roots

• If it never touches the x-axis ⇒ no real roots

Roots of a quadratic equation are simply the x-intercepts of the parabola.

 

3. Solving a Quadratic Equation

3.1 Method 1: Solving by Factorisation

The idea of solving by factorusation is to rewrite the quadratic as a product of two linear factors and then use the zero product property.

Zero Product Property: if A × B = 0, then either A = 0 or B = 0.

 

The Split-the-Middle-Term Technique

For ax² + bx + c = 0:

• Find two numbers p and q such that p + q = b and p × q = a × c

• Rewrite the middle term using p and q

• Factorise by grouping

 

Example 1: Solve 2x² − 5x + 3 = 0

Solution: Here, a = 2, b = −5, c = 3. So a × c = 6.

We need two numbers that add to −5 and multiply to 6. That's −2 and −3.

2x² − 2x − 3x + 3 = 0

2x(x − 1) − 3(x − 1) = 0

⇒ (2x − 3)(x − 1) = 0

So: x = 3/2 or x = 1 

Example 2: Find two consecutive positive integers whose product is 306.

Solution: Let the integers be x and x + 1.

x(x + 1) = 306

⇒ x² + x − 306 = 0

Find two numbers with sum = 1 and product = −306: 18 and −17

x² + 18x − 17x − 306 = 0

x(x + 18) − 17(x + 18) = 0

⇒ (x + 18)(x − 17) = 0

⇒ x = −18 (rejected ∵ negative) or x = 17

The integers are 17 and 18.

3.2 Method 2: Completing the Square

We transform the equation so that one side becomes a perfect square trinomial of the form (x + k)².

Steps: 

• Write the equation in standard form

• Move the constant to the right side

• If a ≠ 1, divide throughout by a

• Add (half of the coefficient of x)² to both sides

• Write the left side as a perfect square

• Take the square root of both sides

• Solve for x

 

Example 1: Solve x² + 4x − 5 = 0

Step 1: x² + 4x = 5

Step 2: Add (4/2)² = 4 to both sides

x² + 4x + 4 = 5 + 4

(x + 2)² = 9

x + 2 = ±3

x = 1 or x = −5

3.3 Method 3: The Quadratic Formula

It directly gives you both roots of any quadratic equation.

The Formula: 

For ax² + bx + c = 0 (where a ≠ 0):

x=−b±b2−4ac2a

 

That ± means you calculate two values:

 

x₁ = (−b + √(b² − 4ac)) / 2a

x₂ = (−b − √(b² − 4ac)) / 2a

 

Example 1: Solve x² − 5x + 6 = 0

a = 1, b = −5, c = 6

Discriminant = (−5)² − 4(1)(6) = 25 − 24 = 1

x = (5 ± √1) / 2 = (5 ± 1) / 2

x = 3 or x = 2

4.The Discriminant

The discriminant, denoted by D (or sometimes Δ).

D = b² − 4ac

The discriminant tells you exactly what kind of roots to expect.

The Three Cases: 

 

Case 1: D > 0 (Positive Discriminant)

The equation has two distinct real roots.

The square root of a positive number is a real number, giving you two different values.

Case 2: D = 0 (Zero Discriminant)

The equation has two equal real roots (also called a repeated root).

Both roots are the same: x = −b/2a

Case 3: D < 0 (Negative Discriminant)

The equation has no real roots.

You'd be taking the square root of a negative number, which has no real solution. 

 

Example 1: Find the nature of roots of x² − 4x + 4 = 0

D = (−4)² − 4(1)(4) = 16 − 16 = 0

⇒ Two equal real roots. Both roots = −(−4)/2(1) = 2

Example 2: Find the nature of roots of 3x² − 5x + 2 = 0

D = (−5)² − 4(3)(2) = 25 − 24 = 1 > 0

⇒ Two distinct real roots

Example 3: Find the nature of roots of x² + x + 1 = 0

D = (1)² − 4(1)(1) = 1 − 4 = −3 < 0

⇒ No real roots

5. Sum and Product of Roots

If α (alpha) and β (beta) are the two roots of ax² + bx + c = 0, then:

 

Sum of roots: α + β = −b/a

Product of roots: α × β = c/a

 

Example: If one root of x² − 5x + k = 0 is 2, find k.

Using sum of roots: 2 + β = 5, so β = 3

Using product of roots: 2 × 3 = k ⇒ k = 6

6. Important Formulas: Quick Reference Card

Click below to download your free Class 10 Maths Chapter 4: Quadratic Equations PDF Notes perfect for last-minute CBSE board exam revision.

Class 10 Maths Chapter 4: Quadratic Equations PDF Notes