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Quadratic Equations Class 10th Notes - Free NCERT Class 10 Maths Chapter 4 Notes - Download PDF

Quadratic Equations Class 10th Notes - Free NCERT Class 10 Maths Chapter 4 Notes - Download PDF

Updated on Apr 19, 2025 12:06 PM IST

Imagine after building your dream house, you need to fence that house, and for that, you need the length and breadth of that house to find the biggest possible area. Quadratic equations can solve this problem. It is an integral part of Mathematics that deals with algebraic equations of degree 2. These mathematical equations function beyond theory because they serve various purposes in real-life scenarios. People use quadratic equations in all parts of life to determine areas and moving object speeds, and make financial predictions while resolving problems throughout physics and engineering. This section creates the fundamental base required to develop algebraic reasoning abilities together with problem-solving capability. The ability to identify and solve quadratic equations will help to succeed in the examinations and ensure success in their subsequent educational and competitive examination programs.

This Story also Contains
  1. Quadratic Polynomial
  2. Quadratic Equation
  3. Roots of a Quadratic Equation
  4. Methods to Solve Quadratic Equations
  5. Nature of Roots
  6. Solving using the quadratic formula when D > 0:
  7. Class 10 Chapter Wise Notes
  8. NCERT Exemplar Solutions for Class 10
  9. NCERT Solutions for Class 10
  10. NCERT Books and Syllabus
Quadratic Equations Class 10th Notes - Free NCERT Class 10 Maths Chapter 4 Notes - Download PDF
Quadratic Equations Class 10th Notes - Free NCERT Class 10 Maths Chapter 4 Notes - Download PDF

The NCERT Class 10 Maths Chapter 4 notes cover a brief outline of the chapter on quadratic equations and can be used for revision. The main topics covered in the quadratic equations Class 10 notes give you the properties and roots of quadratic equations. Class 10 Maths chapter 4 notes also cover the basic equations in the chapter. Quadratic equations Class 10 notes PDF download contains all of these topics. Apart from this, the use of NCERT class 10th maths notes provides you with the ability to examine mathematical concepts rigorously. Also, after studying the NCERT Notes, you will have access to short summaries that help you to remember information fast.

Background wave

Quadratic Polynomial

A polynomial with degree 2 is a quadratic polynomial. It is in the form of: f(x) = ax2 + bx + c, where a ≠ 0.

Quadratic Equation

A quadratic polynomial, when equated to a constant (like 0), then the equation becomes a Quadratic Equation, which means f(x) = 0.

The standard form of a Quadratic Equation is: f(x) = ax2 + bx + c = 0, where a, b and c are the real numbers and a ≠ 0.
Here, a is called as quadratic coefficient, as it is the coefficient of x2 and b is called as linear coefficient, as it is the coefficient of x, and c is the constant term.

Roots of a Quadratic Equation

A quadratic equation's roots represent x values that satisfy the given equation.
Let x = α, and α is a real number. If α satisfies the quadratic equation ax2+ bx + c = 0 such that aα2 + bα + c = 0, then α is the root of the Quadratic Equation.

  • As quadratic polynomials have degree two, quadratic equations can have two roots. Thus, the zeros of a quadratic polynomial f(x) = ax2 + bx + c are the same as the roots of the quadratic equation ax2 + bx + c = 0.
  • Roots can be of three types, in case of a quadratic equation: two distinct roots, two equal roots or real roots may not exist.

Methods to Solve Quadratic Equations

There are three methods to solve Quadratic Equations.

1. Factorization Method

In this method, factorize the equation into two linear factors and equate each factor to zero to find the roots of the equation.

Step 1: Quadratic Equation in the form of ax2 + bx + c = 0.

Step 2: Find the two numbers (let's say p and q), whose sum is equal to 'b' and product is equal to 'a × c'.

Step 3: By factorization, write ax2 + bx + c = 0 as (x + p) (x + q) = 0

For example-

x2- 2x - 15=0

⇒ (x+3)(x-5)=0

So, x + 3 = 0 or, x - 5 = 0

x = - 3 or x = 5

The above values of x are the two roots of the given quadratic equation.

2. Completing the Square Method

In this method, convert the equation to square form (x + a)2 - b2 = 0 or (x + a)2 = b2 to find the roots.

Example: Solve x2+6x+5=0

Solution:

  • Step 1: Make sure the coefficient of x² is 1
    In this case, it's already 1.
    If not, divide the equation with the coefficient of x².

  • Step 2: Move the constant term to the right-hand side:
    x2+6x=5

  • Step 3: Add the square of half the coefficient of x to both sides:
    Half of 6 is 3, and 32=9
    So, x2+6x+9=5+9

  • Step 4: Write the left-hand side as a perfect square, to make the form (x + a)2 = b2
    (x+3)2=4

  • Step 5: Take the square root of both sides:
    x+3=±4

  • Step 6: Solve for x:
    x+3=±2
    x=3±2

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Therefore, x = -1 or -5

3. Quadratic formula method

In this method, find the roots by using the quadratic formula. The quadratic formula is

x=b±b24ac2a

where a, b and c are the real numbers and b2 – 4ac is known as the discriminant. In this method, we directly get the roots once we substitute the values in the formula.

Example: Solve 2x24x6=0

Solution: Here, a = 2, b = -4, c = -6
Apply the quadratic formula: x=b±b24ac2a

x=(4)±(4)24(2)(6)2(2)

x=4±16+484

x=4±644

x=4±84

x=4+84=3orx=484=1

x=3orx=1

Nature of Roots

The nature of the roots of the equation depends upon the value of D, which is called the discriminant.

Value of discriminant

Number of roots

D > 0

Two distinct real roots

D = 0

Two equal and real roots

D < 0

No real roots


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Solving using the quadratic formula when D > 0:

When the quadratic equation has D > 0, the roots of the equation are distinct and real.

Example: Solve x25x+6=0

Solution: Here, a = 1, b = -5, c = 6

D = (5)24(1)(6) = 25 - 24 = 1

It means D > 0.

Now, apply the quadratic formula:

x = (5)±12(1)=5±12

x = 3orx = 2

Therefore, the roots are distinct and real.

Solving using the quadratic formula when D = 0:

When the quadratic equation has D = 0, the roots of the equation are equal.

Example: Solve x24x+4=0

Solution: Here, a = 1, b = -4, c = 4

D = (4)24(1)(4)= 16 - 16 = 0

It means D = 0.

Now, apply the quadratic formula:

x = (4)2(1)=42 = 2

Therefore, x = 2

Thus, the roots are equal, as we get only one root.

Solving using the quadratic formula when D < 0:

When the quadratic equation has D < 0, the roots of the equation are not real (complex).

Example: Solve x2+2x+5=0

Solution: Here, a = 1, b = 2, c = 5

D = (2)24(1)(5) = 4 - 20 = -16

It means D < 0.

Now, apply the quadratic formula:

x = 2±162(1)=2±4i2

x = 1±2i

Thus, the roots are not real and are complex numbers.

Class 10 Chapter Wise Notes

Students must download the notes below for each chapter to ace the topics.


NCERT Exemplar Solutions for Class 10

Students must check the NCERT Exemplar solutions for class 10 of Mathematics and Science Subjects.

NCERT Solutions for Class 10

Students must check the NCERT solutions for class 10 of Mathematics and Science Subjects.

NCERT Books and Syllabus

To learn about the NCERT books and syllabus, read the following articles and get a direct link to download them.

Frequently Asked Questions (FAQs)

1. What is a quadratic equation?

A quadratic polynomial, when equated to 0, then the equation becomes a Quadratic Equation, which means f(x) = 0.

The standard form of a Quadratic Equation is: f(x) = ax+ bx + c = 0, where a, b and c are the real numbers and a ≠ 0.

2. How do you identify a quadratic equation?

To recognize a quadratic equation, check:

  •  It is a polynomial equation with one variable.
  •  The variable's highest exponent is two.
  •  The leading coefficient (a) must not be 0.
3. What are the different methods to solve quadratic equations?

The methods for solving quadratic equations are as follows:

  • Factorization Method
  • Completing the Square Method
  • Quadratic Formula Method
4. What is the standard form of a quadratic equation?

The standard form of a Quadratic Equation is: f(x) = ax+ bx + c = 0, where a, b and c are the real numbers and a ≠ 0.

5. What is the significance of the discriminant in a quadratic equation?

The nature of the roots of the equation depends upon the value of D, which is called the discriminant, where b– 4ac is known as the discriminant.

It tells if:

  • D > 0: Two distinct real roots
  • D = 0: Two equal real roots
  • D < 0: No real roots (roots are complex)

Articles

A block of mass 0.50 kg is moving with a speed of 2.00 ms-1 on a smooth surface. It strikes another mass of 1.00 kg and then they move together as a single body. The energy loss during the collision is

Option 1)

0.34\; J

Option 2)

0.16\; J

Option 3)

1.00\; J

Option 4)

0.67\; J

A person trying to lose weight by burning fat lifts a mass of 10 kg upto a height of 1 m 1000 times.  Assume that the potential energy lost each time he lowers the mass is dissipated.  How much fat will he use up considering the work done only when the weight is lifted up ?  Fat supplies 3.8×107 J of energy per kg which is converted to mechanical energy with a 20% efficiency rate.  Take g = 9.8 ms−2 :

Option 1)

2.45×10−3 kg

Option 2)

 6.45×10−3 kg

Option 3)

 9.89×10−3 kg

Option 4)

12.89×10−3 kg

 

An athlete in the olympic games covers a distance of 100 m in 10 s. His kinetic energy can be estimated to be in the range

Option 1)

2,000 \; J - 5,000\; J

Option 2)

200 \, \, J - 500 \, \, J

Option 3)

2\times 10^{5}J-3\times 10^{5}J

Option 4)

20,000 \, \, J - 50,000 \, \, J

A particle is projected at 600   to the horizontal with a kinetic energy K. The kinetic energy at the highest point

Option 1)

K/2\,

Option 2)

\; K\;

Option 3)

zero\;

Option 4)

K/4

In the reaction,

2Al_{(s)}+6HCL_{(aq)}\rightarrow 2Al^{3+}\, _{(aq)}+6Cl^{-}\, _{(aq)}+3H_{2(g)}

Option 1)

11.2\, L\, H_{2(g)}  at STP  is produced for every mole HCL_{(aq)}  consumed

Option 2)

6L\, HCl_{(aq)}  is consumed for ever 3L\, H_{2(g)}      produced

Option 3)

33.6 L\, H_{2(g)} is produced regardless of temperature and pressure for every mole Al that reacts

Option 4)

67.2\, L\, H_{2(g)} at STP is produced for every mole Al that reacts .

How many moles of magnesium phosphate, Mg_{3}(PO_{4})_{2} will contain 0.25 mole of oxygen atoms?

Option 1)

0.02

Option 2)

3.125 × 10-2

Option 3)

1.25 × 10-2

Option 4)

2.5 × 10-2

If we consider that 1/6, in place of 1/12, mass of carbon atom is taken to be the relative atomic mass unit, the mass of one mole of a substance will

Option 1)

decrease twice

Option 2)

increase two fold

Option 3)

remain unchanged

Option 4)

be a function of the molecular mass of the substance.

With increase of temperature, which of these changes?

Option 1)

Molality

Option 2)

Weight fraction of solute

Option 3)

Fraction of solute present in water

Option 4)

Mole fraction.

Number of atoms in 558.5 gram Fe (at. wt.of Fe = 55.85 g mol-1) is

Option 1)

twice that in 60 g carbon

Option 2)

6.023 × 1022

Option 3)

half that in 8 g He

Option 4)

558.5 × 6.023 × 1023

A pulley of radius 2 m is rotated about its axis by a force F = (20t - 5t2) newton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is 10 kg m2 , the number of rotations made by the pulley before its direction of motion if reversed, is

Option 1)

less than 3

Option 2)

more than 3 but less than 6

Option 3)

more than 6 but less than 9

Option 4)

more than 9

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