Class 10 Math Quadratic Equations Notes 2025 & Study Material PDF

Mathematics remains an important part of the world since students will learn how to analyse and solve problems as civilians. One of the complex concepts pupils grapple with in Class 10 Math Quadratic Equations Notes. This is the building block for further algebra, calculus, and a multitude of other real-world applications. In simple terms, knowing how to solve quadratic equations is a skill that provides one with a deeper comprehension of how numerous mathematical functions model real-world situations. If you happen to be a teacher looking for premium Class 10 quadratic equations notes, this guide is meant for you because it is tailored to make the process of learning as effective as possible. Get your quality study materials such as worksheets and solved examples from Educart and broaden your horizons.

CBSE Class X Quadratic Equation Notes

Class 10 math Chapter 4 notes cover all the main concepts like substitution method and solving linear equations. Quadratic Equation includes every number other than the complex numbers and the downloadable notes PDFs provided below are detailed and in easy-to-understand language.

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Class 10 Quadratic Equations: Notes

The quadratic equations will be helpful for the exams.

What is a Quadratic Equation?

A quadratic equation is any equation that can be written in the form:

ax²+bx+c=0

where:

• x represents an unknown variable,
• a, b, and c are constants (numbers),
• and aaa cannot be zero (if a=0, then the equation is linear, not quadratic).

In simple terms, it's an equation that involves x² (the square of x) and may also include a term with just x, as well as a constant number.

Examples:

2x²+5x−3=0

x²−4x+4=0

3x²−2x+1=0

These are all quadratic equations.

Important Features of Quadratic Equations:

The highest power of x in a quadratic equation is always 2. That’s why it's called “quadratic” (from "quadratus" meaning "square").

The graph of a quadratic equation is a parabola. This means it has a U-shape (either opening upward or downward).

Solving Quadratic Equations

There are several methods to solve quadratic equations, including:

• Factoring
• Completing the Square
• Using the Quadratic Formula
• Graphing

Let’s look at each method in detail.

Factoring Method:

Factoring is one of the simplest methods, but it only works when the quadratic equation can be easily factorised.

Steps:

• Find two numbers that multiply to give ac (the product of a and c) and add to give b (the coefficient of x).
• Split the middle term into two terms using those two numbers.
• Factor by grouping. Group the terms in pairs and factor each group.
• Solve for x by setting each factor equal to zero and solving for the variable.

Example of Factoring:

Consider the equation:

x²+5x+6=0

• Multiply a and c: Here, a = 1 and c = 6
• Find two numbers that multiply to 6 and add up to 5 (the middle term). These numbers are 2 and 3.
• Split the middle term (5x) into 2x and 3x:

x²+2x+3x+6=0

• Factor by grouping: Group the terms as follows:

(x²+2x)+(3x+6)=0

Now factor each group:

x(x+2)+3(x+2)=0

• Factor out the common binomial:

(x+2)(x+3)=0

• Solve for x: Set each factor equal to zero:

x+2=0 or x+3=0

So, x=−2 or x=−3.

Solutions: x = -2 and x=−3.

Completing the Square:

This method is useful when factoring is difficult or impossible. It involves transforming the equation into a perfect square trinomial.

Steps:

• Start with the equation in the form ax²+bx+c=0.
• If a≠1, divide through by a to make the coefficient of x² equal to 1.
• Move the constant term to the other side of the equation.
• Add a specific value to both sides to make the left-hand side a perfect square trinomial.
• Factor the left-hand side and solve for x.

Example of Completing the Square:

Consider the equation:

x²+6x−7=0

• Move the constant term to the other side:

x²+6x=7

• Add (6/2)²=9 to both sides:

x²+6x+9=7+9

x²+6x+9=16

• Factor the left-hand side:

(x+3)²=16

• Solve for x by taking the square root of both sides:

x+3=±4

So,

x+3=4 or x+3=−4

Therefore, x=1or x=−7.

Solutions: x=1 and x=−7

Quadratic Formula:

The quadratic formula is a universal method that 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,
• ± means there will be two solutions (one for the plus and one for the minus).

Steps:

• Identify a, b, and c in the equation.
• Substitute these values into the quadratic formula.
• Simplify under the square root (the discriminant), and then calculate the two possible values for x.

Example of the Quadratic Formula:

For the equation:

x²+4x−5=0

• Identify a=1, b=4, and c=−5.
• Plug these values into the formula:

x= (−4±(√4²-4(1)(-5)))/2(1)

• Simplify the expression under the square root:

x= (−4±(√16+20)/2

x= (−4±(√36)/2

• Take the square root of 36:

x= (−4± 6)/2

• Now solve for the two possible values of x:

X = (−4+ 6)/2 = 2/2 =1

x = (−4- 6)/2 = -10/2 = -5

Solutions: x=1 and x=−5

Graphing Quadratic Equations:

The graph of a quadratic equation is a parabola. The general shape of the graph depends on the sign of a:

• If a>0, the parabola opens upwards (like a "U").
• If a<0, the parabola opens downward (like an "n").

The vertex of the parabola is the highest or lowest point (depending on the direction it opens), and the axis of symmetry is a vertical line that passes through the vertex.

To find the vertex of the parabola for the equation ax²+bx+c=0, use the formula:

x=-b/2a

Once you find the x-coordinate of the vertex, substitute it back into the equation to find the y-coordinate.

Class 10 Quadratic Equations: Important Questions 

Here are some important questions on Quadratic Equations for Class 10, covering a range

of problem types that help students to prepare well for their exams:

Basic Solving Using Factoring

Q1: Solve the quadratic equation by factoring:

x²−7x+12=0

Solution:

• Find two numbers that multiply to +12 and add up to −7. The numbers are −3 and −4.
• Factor the quadratic:

(x−3)(x−4)=0

• Solve for x:

x=3 or x=4

Answer: x=3,4

Solving Using the Quadratic Formula

Q2: Solve the quadratic equation using the quadratic formula:

2x²+3x−5=0

Solution:

• The quadratic formula is x=(−b±(√b²−4ac)/2a)
• Here, a=2, b=3, and c=−5.
• Substitute into the formula:

x= −3±√3²−4(2)(-5))/2(2))

x= (−3±√9+40)/4

x= (−3±√49)/4

x= (−3±7)/4

x=(−3±7​)/4

So, x=−3+7/4=1

 or x=−3−7/4=−2.5

Answer: x=1,−2.5

Completing the Square

Q3: Solve the quadratic equation by completing the square:

x²+6x−7=0

Solution:

• Move the constant term to the other side:

x²+6x=7

• Take half of the coefficient of x, square it, and add it to both sides:

(6/2​)² =9

Add 9 to both sides:

x²+6x+9=7+9

(x+3)²=16

Take the square root of both sides:

x+3=±4

Solve for x:

x=−3+4=1 or x=−3−4=−7

Answer: x=1,−7

 Word Problem Involving Quadratic Equations

Q4: A rectangular garden has a length of 2 m², more than its width. If the area of the garden is 120 m², find the dimensions of the garden.

Solution:

• Let the width be x meters.
• The length will be x+2 meters.
• Area of the rectangle is given by length×width=120.

So, we have the equation:

x(x+2)=120

x²+2x=120

x²+2x−120=0

Now, solve the quadratic equation by factoring:

We need two numbers that multiply to −120 and add to 2. These numbers are 12 and −10.

Factor the equation:

(x+12)(x−10)=0

Solve for xxx:

x=−12 (not valid as width can't be negative) or x=10

So, the width is 10 m, and the length is 10+2=12 m.

Answer: The width is 10 m, and the length is 12 m.

Nature of Roots Using the Discriminant

Q5: Find the nature of the roots of the quadratic equation:
3x²−5x+2=0

Solution:

• The discriminant D=b²−4ac
• For the equation 3x2−5x+2=0, a=3, b=−5, and c=2.

Discriminant:

D=(−5)²−4(3)(2)

=25−24

=1

Since the discriminant is positive and greater than 0, the equation has two real and distinct roots.

Answer: Two real and distinct roots.

Solving a Quadratic Equation with Complex Roots

Q6: Solve the quadratic equation:

x²+2x+5=0

Solution:

• For x²+2x+5=0, use the quadratic formula:

x= −(2±√2²- 4 (1)(5))/2(1)

x= (−2±√4 -20)/2

x= (−2±√-16)/2

Since the discriminant is negative, the roots will be complex. Simplifying further:

x= −2±4i​/2

x=−1±2i

Answer: The solutions are x=−1+2i and x=−1−2

These questions cover a wide range of topics related to quadratic equations, including solving

using different methods (factoring, quadratic formula, completing the square), interpreting word

problems, and understanding the nature of roots. Practicing these will help build a strong

foundation for the Class 10 exams!

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Achieving a Better Grade in Quadratic Equations - Class 10

Quadratic equations are one of the most important topics covered in Class 10 Mathematics and are quite important in algebra. Subsequently, their applications are numerous: in Maths, engineering, finance, computer science, etc. To do well in this chapter, students ought to pay attention to the concepts, do ample practice, and develop good examination techniques. Thus, here are some tips to help you improve on your Quadratic Equations chapter:

Learn All Topics in Depth

Have a clear picture of how you will approach solving the problems. Important concepts include knowing the definition and standard form of quadratic equations, quadratics, and the discriminants and roots (-b +/- 2 square root b2 - 4ac).

Learn and Practice Quadratic Equation Methods

A quadratic equation can be solved in 3 different ways:

1. Factorisation—splitting the middle term

2. Quadratic Formula: x = (-b ± (b² - 4a)¹/²) / 2a

3. Completing the Square—Multiplying the quadratic in standard form until it reaches perfect square form

Solve as many different types of problems:

Put in enough work practising textbook questions, old exam question papers, and extra workbooks. Be sure to include the word problems too because those tend to show up more often, and they are practical-based and need application.

Make Use of Quick Revision Notes

Make sure you prepare short notes that contain formulas and relevant key concepts with solved examples. By having short notes and flashcards.

Use of Quadratic Equations in Real-Life

Like many other aspects of studies, quadric equations have importance in other fields such as:

Maths as well as Engineering:

Motion of Projectiles: The route that a ball takes when it is thrown into the air can be described using a quadratic function.

Business and Economics: 

Estimating Revenues or Profits: A business does market research into the gain and loss profitability, and this includes procedures of calculations of a quadratic function. 

Civil Engineering and Architecture: 

Bridges and Arches: Engineers design ornate yet simple curves with the aid of quadratic equations. 

Biology and Medicine: 

Modelling Heart Rate: In medicine, the efficient computation of biological and medical data needs the adoption of quadratic equations.

Achieving a Better Grade in Quadratic Equations - Class 10

Quadratic equations are one of the most important topics covered in Class 10 Mathematics and are quite important in algebra. Subsequently, their applications are numerous: in Maths, engineering, finance, computer science, etc. To do well in this chapter, students ought to pay attention to the concepts, do ample practice, and develop good examination techniques. Thus, here are some tips to help you improve on your Quadratic Equations chapter:

Learn All Topics in Depth

Have a clear picture of how you will approach solving the problems. Important concepts include knowing the definition and standard form of quadratic equations, quadratics, and the discriminants and roots (-b +/- 2 square root b2 - 4ac).

Learn and Practice Quadratic Equation Methods

A quadratic equation can be solved in 3 different ways:

1. Factorisation—splitting the middle term

2. Quadratic Formula: x = (-b ± (b² - 4a)¹/²) / 2a

3. Completing the Square—Multiplying the quadratic in standard form until it reaches perfect square form

Solve as many different types of problems:

Put in enough work practising textbook questions, old exam question papers, and extra workbooks. Be sure to include the word problems too because those tend to show up more often, and they are practical-based and need application.

Make Use of Quick Revision Notes

Make sure you prepare short notes that contain formulas and relevant key concepts with solved examples. By having short notes and flashcards.

Where Can You Get Quadratic Equations Notes for Download?

Educational platforms provide complete quadratic equations notes question banks and worksheets for students as well as teachers, organised by chapters.

For students of Class 10, quadratic equations are indeed useful in mathematics and relevant in other domains and even life. Organised learning resources, question banks, and supporting worksheets promote positive mathematical skills. It is useful to have comprehensive notes on quadratic equations, no matter if it is the teachers who are preparing the lessons or students trying to excel in their exams. Without any scepticism, improve your skills! Grab your Class 10 Quadratic Equations Notes now!