CBSE Notes Class 10 Maths Chapter 4 Quadratic Equations

CBSE Class 10 Chapter 4 focuses on Quadratic Equations. Quadratic equations are one of the main concepts of algebra, and they are applied to many fields. Quadratic equations are equations of the form $\mathrm{a}{x}^2+\mathrm{b}x+\mathrm{c}=0$, where a, b, and c are constants, and $a\ne 0$. This chapter deals with methods for solving quadratic equations, understanding the nature of their roots, and applying these concepts in solving real-world problems. It is expected that these sets of notes will help to give a structured understanding regarding quadratic equations, simplifying complex theories and streamlining the preparation for an exam.

1.0CBSE Class 10 Maths Chapter 4 Quadratic Equations - Revision Notes

Important Concepts in Quadratic Equations:

• Standard Form of Quadratic Equations: Any equation in the form $a{x}^2+bx+c=0$ is a quadratic equation where a, b, and c are constants. 
• Roots of a Quadratic Equation: The values of x that satisfy the equation$a{x}^2+bx+c=0$ are called the roots or solutions of the quadratic equation.
• Nature of Roots: The discriminant (D = b²− 4ac ) determines the nature of the roots:

(a). If D > 0: two distinct real roots.

(b). If D = 0: two equal real roots.

(c). If D < 0: two complex roots (no real solutions).

1. Methods of Solving Quadratic Equations:

• Factorization Method: Involves breaking down the quadratic equation into two linear factors and solving for 𝑥.
• Completing the Square: Rewrites the equation as a perfect square and isolates 𝑥 to solve.
• Quadratic Formula: Directly solves the equation using the formula $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

2. Definitions:

• Quadratic Equation: An equation in the form a𝑥² + b𝑥 + c = 0 with a ≠  0.
• Discriminant (D): The expression b² – 4ac = 0 helps determine the nature of the roots.
• Roots or Solutions: Values of 𝑥 that satisfy the equation. In quadratics, each equation has two roots, which can be real or complex.
• Vertex of Parabola: The turning point of the graph of a quadratic function, which represents the maximum or minimum point of the parabola.

3. Formulas:

•  Quadratic Formula: $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$
• Discriminant (D): $D=b^2-4ac$
• The sum of Roots: $\alpha +\beta =-b/a$
• Product of roots: $\alpha \cdot \beta =c/a$

4. Tips and Tricks:

• Identify Solving Technique: If the equation will easily factor, use the factorisation method. Otherwise, use the formula for roots directly since they produce quicker answers than the quadratic formula.
• Remembering the Discriminant: Keep in mind that the discriminant b² – 4ac, determines if roots are real or complex.
• Shortcut for Sum and Product: Use the formulas for the sum & product of roots for quick calculations in multiple-choice questions.
• Graphical Representation: It is easier to comprehend the characteristics of the roots and the vertex when the quadratic function is visualised as a parabola.

2.0Solved Problems

Problem 1:

Create a quadratic equation to represent the following scenario: A rectangular plot has an area of 528 m². The plot is more than twice as long (in meters) as it is wide. 

Solutions:

Let us consider,

The breadth of the rectangular plot = x m

Thus, the length of the plot = (2x + 1) m

As we know,

Area of rectangle = length × breadth = 528 m²

When we enter the plot's length and width values into the formula, we obtain,

(2x + 1) × x = 528

⇒ 2x² + x =528

⇒ 2x² + x – 528 = 0

As a result, the plot's length and width fulfil the quadratic equation, 2x² + x - 528 = 0, which is the mathematical description of the problem.

Problem 2: Find two consecutive positive integers, the sum of whose squares is 365.

Solution:

Let us say the two consecutive positive integers are x and x + 1.

Therefore, as per the given questions,

x² + (x + 1)² = 365

⇒ x² + x² + 1 + 2x = 365

⇒ 2x² + 2x – 364 = 0

⇒ x² + x – 182 = 0

⇒ x² + 14x – 13x – 182 = 0

⇒ x (x + 14) -13 (x + 14) = 0

⇒ (x + 14) (x – 13) = 0

Thus, either, x + 14 = 0 or x – 13 = 0,

⇒ x = – 14 or x = 13

Since the integers are positive, x can be 13 only.

∴ x + 1 = 13 + 1 = 14

3.0Key Features of CBSE Maths Notes for Class 10 Chapter 4

• Clear Explanations & Simplified Language: These notes make it easier for students to follow the stages involved in solving quadratic equations by presenting difficult topics in plain, understandable language.
• Solved Examples: Strategy at each stage of solving quadratic equations. This helps the students understand how it pertains to many types of issues.
• Tables & Formula Summaries: Critical formulas, such as the quadratic formula and criteria for a discriminant, are summarised in tables so that students may readily memorise and refer to them.
• Diagrams & Visual Aids: Graphs and diagrams of parabolas present the nature of quadratic functions in behaviour with roots, vertex, and shapes in the graph, hence making concepts easier to visualise and remember.

These CBSE Maths Notes for Class 10 Chapter 4 on Quadratic Equations help students learn to solve quadratic equations in detail. These notes facilitate students' comprehension and successful solution of quadratic equations by providing concise explanations, solved examples, and useful review advice.