CBSE Notes Class 10 Maths Chapter 4 Quadratic Equations

Maths Chapter 4 in CBSE Class 10 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 CBSE Notes will help to give a structured understanding regarding quadratic equations, simplifying complex theories and streamlining the preparation for an exam.

1.0Download CBSE Notes for Class 10 Maths Chapter 4: Quadratic Equations - Free PDF!!

To easily grasp the concepts of Quadratic Equations, CBSE Notes for Class 10 students can download free notes for the 4th chapter. This readily accessible PDF offers a clear and understandable explanation.

2.0CBSE 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.

3.0CBSE Class 10 Maths Chapter 4 Quadratic Equations - Key Notes

A quadratic equation in the variable x is an equation of the form ax² + bx + c = 0, where a, b, c are real numbers, a ≠ 0.

e.g., 2x² – 3x + 1 = 0, 4x – 3x² = 0, 1 – x² = 0 etc.

Roots of quadratic equation :

x = α is said to be root of the quadratic equation ax² + bx + c = 0, a ≠ 0, if x = α satisfies the quadratic equation, i.e. in other words, the value of aα² + bα + c is zero.

Solving Quadratic Equations through Factorisation

Given a quadratic equation ax² + by + c = 0, a ≠ 0.

Algorithm

Step-I : Factorise the product of constant term and coefficient of x² of the given quadratic equation.

Step-II : Express the coefficient of middle term as the sum or difference of the factors obtained in step – I. Clearly, the product of these two factors will be equal to the product of the coefficient of x² and constant term.

Step-III : Split the middle term in two parts obtained in step – II.

Step-IV : Factorise the quadratic equation obtained in step – III by grouping method.

Solving Quadratic Equations through Quadratic Formula

Discriminant :

If ax² + bx + c = 0, a ≠ 0 (a, b, c ∈ R) is a quadratic equation, then the expression b² – 4ac is known as its discriminant and is generally denoted by D or Δ.

Consider quadratic equation ax² + bx + c = 0, a ≠ 0,

then,

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

The roots of the equation are

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

or

$\frac{-b-\sqrt{b^2-4ac}}{2a}$

$x=\frac{-b+\sqrt{D}}{2a}\text{or}x=\frac{-b-\sqrt{D}}{2a},\text{ where }D=b^2-4ac.$

Nature of roots

Let the quadratic equation be ax² + bx + c = 0, where a ≠ 0 and a, b, c ∈ R, if α and β are two roots of the quadratic equation.

Then,

$$\begin{gathered} \alpha =\frac{-b+\sqrt{D}}{2a}\text{and} \\ \beta =\frac{-b-\sqrt{D}}{2a} \end{gathered}$$

$$\begin{gathered} \text{Discriminant}(D=b^2-4ac) \\ IfD\ge 0\to \text{Real Roots} \\ IfD<0\to \text{Imaginary or No Real Roots} \\ IfD>0\to \text{Real and distinct} \\ \alpha =\frac{-b+\sqrt{D}}{2a} \\ \beta =\frac{-b-\sqrt{D}}{2a} \\ IfD=0\to \text{Real and equal} \\ \alpha =-\frac{b}{2a}=\beta \end{gathered}$$

Formation of quadratic equations using sum & product

Let α and β be the roots of the quadratic equation ax² + bx + c = 0, a ≠ 0.

Then

$$\begin{gathered} \alpha =\frac{-b+\sqrt{b^2-4ac}}{2a}\text{and} \\ \beta =\frac{-b-\sqrt{b^2-4ac}}{2a} \end{gathered}$$

therefore The sum of roots

$$\begin{gathered} \alpha +\beta =-\frac{b}{a}=\frac{\text{coefficient of }x}{\text{coefficient of }{x}^2} \\ \text{and product of roots} \\ \alpha \beta =\frac{c}{a}=\frac{\text{constant term}}{\text{coefficient of }{x}^2} \\ \text{Hence the quadratic equation whose roots are}\alpha and\beta isgivenby \\ {x}^2-(\alpha +\beta )x+\alpha \beta =0 \end{gathered}$$

Problems on Numbers and Ages

Step-I : Translating the word problem into symbolic language (mathematical statement) which means identifying relationships existing in the problem and then forming the quadratic equation.

Step-II : Solving the quadratic equation thus formed.

Step-III : Interpreting the solution of the equation which means translating the result of mathematical statement into verbal language.

Note : If a is unit digit of a number and b is ten's digit of a number then the number is 10b + a.

Problems on geometrical concepts

1. Sum of angles of a triangle is 180°.
2. Sum of angles of a quadrilateral is 360°.
3. Area of triangle = ½ × base × height
4. In ΔABC right angled at B, AB² + BC² = AC²

Problems on time and distance

1. Distance = speed × time
2. 1 hour = 60 minutes
3. Let x km/hr be the speed of boat in still water and let y km/hr be speed of stream.

Then speed upstream = (x − y) km/hr
Speed downstream = (x + y) km/hr

Problems on time and work

If A alone takes x days to complete some work. Then, A's 1-day work = 1/x

Solution by completing the square

Step-I : Obtain the quadratic equation. Let the quadratic equation be ax² + bx + c = 0, a ≠ 0.

Step-II : Make the coefficient of x² unity by dividing throughout by it, if it is not unity, i.e. obtain

$$\begin{gathered} x^2+\frac{b}{a}x+\frac{c}{a}=0 \\ \text{Step-III :}Shifttheconstantterm\frac{c}{a}onRHStoget \\ {x}^2+\frac{b}{a}x=-\frac{c}{a} \\ \text{Step-IV :}\text{Add square of half of the coefficient of x,} \\ i.e.{\left(\frac{b}{2a}\right)}^{2}onbothsidestoobtain \\ {x}^2+2\left(\frac{b}{2a}\right)x+{\left(\frac{b}{2a}\right)}^2={\left(\frac{b}{2a}\right)}^2-\frac{c}{a} \\ \text{Step-V :}WriteL.H.S.astheperfectsquareandsimplifyRHStoget \\ {\left(x+\frac{b}{2a}\right)}^2=\frac{b^2-4ac}{4a^2} \\ \text{Step-VI :}Takesquarerootofbothsidestoget \\ x+\frac{b}{2a}=\pm \sqrt{\frac{b^2-4ac}{4a^2}} \\ \text{Step-VII :}Obtainthevaluesofxbyshiftingtheconstantterm\frac{b}{2a}onRHS \\ x=-\frac{b}{2a}\pm \sqrt{\frac{b^2-4ac}{4a^2}} \\ x=\frac{-b\pm \sqrt{b^2-4ac}}{2a} \end{gathered}$$

4.0Solved 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

5.0Key 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 NOtes for Class 10 Maths - 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.