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Introduction to Quadratic Equation – Definition, Facts, Formula, Examples | How to Solve Quadratic Equations?

Introduction to Quadratic Equations

A quadratic equation has two solutions that may be or may not be distinct. The result may be real numbers or imaginary numbers. Learn the important formulas of quadratic equation, definition here. Let us learn about the introduction to the quadratic equations from this article. You can find examples of quadratic equations with step-by-step explanations.

What is a Quadratic Equation?

In the name, quadratic “quad” means square because the equation is square. A quadratic equation is an algebraic expression of the 2nd degree in variable x. The variable x has two answers real or complex numbers. The answers or solutions of x are called roots of the quadratic equations. They are specified as (α, β). The standard form of the quadratic equation is ax² + bx + c = 0. Where a, b is the coefficient of x² and c is the constant. a,b, c are not fractions nor decimals.

Quadratic Equation Formula

The formula for the quadratic equation is an easy method to find the roots of the equation. Without the formulas, the values are not factorized, and can find the roots in the easiest way. The roots of Q.E helps to find the sum of the roots and product of the roots of the quadratic equation.
Quadratic Equation (α, β) = [-b ± √(b² – 4ac)]/2a.

Important Formulas to Solve Quadratic Equations

  • The standard form of the quadratic equation is ax² + bx + c = 0.
  • The discriminant(D) of quadratic equation is D = b² – 4ac.
  • For the case, D = 0 the roots are real and equal.
  • For the case, D > 0 the roots are real and distinct.
  • For the case, D < 0 the roots do not exist, or the roots are complex.
  • The product of the Root of the quadratic equation is αβ = c/a = Constant term/ Coefficient of x²
  • The roots of the quadratic equation is x = [-b ± √(b² – 4ac)]/2a.
  • The sum of the roots of a Q.E is α + β = -b/a = – Coefficient of x/ Coefficient of x²
  • Quadratic equation in the form of roots is x² – (α + β)x + (αβ) = 0
  • If α, β, γ are roots of a cubic equation ax³ + bx² + cx + d = 0, then, α + β + γ = -b/a, αβ + βγ + λα = c/a, and αβγ = -d/a
  • The roots (α + iβ), (α – iβ) are the conjugate pair of each other.
  • For a > 0, the quadratic expression f(x) = ax² + bx + c has a minimum value at x = -b/2a
  • For a < 0, the quadratic expression f(x) = ax² + bx + c has a maximum value at x = -b/2a
  • For a > 0, the range of the quadratic equation ax² + bx + c = 0 is [b² – 4ac/4a, ∞).
  • For a < 0, the range of the quadratic equation ax² + bx + c = 0 is (∞, -(b² – 4ac)/4a]

Methods for Solving Quadratic Equations

There are three methods for solving quadratic equations. They are as follows,
1. Factorization method
2. Completing the square method
3. Quadratic Equation formula

Quadratic Equation Question and Answers

Example 1.
Solve 5x² + 7x + 2 = 0
Solution:
Coefficients are: a = 5, b = 7, c = 2
x = [-b ± √(b² – 4ac)]/2a
x = [-7 ± √(7² – 4.5.2)]/2.5
x = [-7 ± √(49 – 40)]/10
x = [-7 ± √(9)]/10
x = [-7+3]/10 = -4/10 = -2/5
x = [-7 – 3]/10 = -10/10 = -1
Thus x = -2/5 or x = -1

Example 2.
Find the range of k for which 4 lies between the roots of the quadratic equation x² + 2(k – 4)x + 5 = 0.
Solution:
6 will lie between the roots of the quadratic expression f(x) = x² + 2(k – 4)x + 5 if,
f(4) < 0
= 16 + 2(k – 4)4 + 5 < 0
= 16 + (2k – 8)4 + 5 < 0
= 16 + 8k – 32 + 5 < 0
= 8k – 11 < 0
= k < 11/8

Example 3.
Find the factors of the quadratic equation x² + 7x + 12 = 0
Solution:
x² + 7x + 12 = 0
x² + 3x + 4x + 12 = 0
x(x + 3) + 4(x + 3) = 0
(x + 4) (x + 3) = 0
x + 4 = 0 or x + 3 = 0
x = -4 or x = -3

FAQs on Quadratic Equation

1. What is the purpose of quadratic equations?

Quadratic equations are actually used in our daily life, as when calculating areas, determining a product’s profit or formulating the speed of an object.

2. What is the standard form of the quadratic equation?

The standard form of the quadratic equation is ax² + bx + c = 0

3. How many roots does a quadratic equation have?

The quadratic equation has two roots. The Q.E with real or complex coefficients has two solutions that are called roots.

Calendar Guides Us to Know – Benefits, Disadvantages, Examples | Importance of Calendar in School

Calendar Guides Us to Know

Most of us understand the value of planning and preparation. The calendar plays an important role in planning a specific day. From a student’s point of view, it is important to remember the exam dates, friends’ birthdays, special dates, etc. Learn how to make use of Calendar guides from here. Students of 4th grade can know the importance of a calendar with the help of this article.

A calendar is a figure that shows days and months. The calendar is one of the most important topics for fixing the exam dates for all government entrance exams. The calendar shows days, months, and years. Let us see how the calendar guides us to know about the special dates, even and odd months, leap and non-leap years, etc.

Importance of Calendar | Benefits of Using a Calendar

i. Total number of Sundays in a month.
ii. Number of days in a month.
iii. Number of months in a year.
iv. To remember a special day or definite day.
v. Different number of days in a month like January, February, April, etc.
v. We can know what is a leap year is and which is not a leap year.
vi. It helps to fix registration dates, competitions, etc.
vii. You can schedule your work whether it is urgent or important tasks with the help of a calendar.
viii. Not urgent and important can be scheduled and added to ‘to-do list’ and urgent and unimportant tasks can be delayed with the help of the calendar.

Wanna become a master in maths? If yes, then learn the fundamentals of maths at the primary level itself. Get the worksheets, practice tests, examples, word problems on 4th Grade Math and get guidance.

Examples on Purpose of a Calendar

Example 1.
How many days are there in February in a leap year?
Solution:
There are 29 days in February in a leap year.

Example 2.
Which is the first day of the year in 2020?
Solution:
The first day of the year 2020 is Wednesday, January 1st.

Example 3.
How many days are there in the month of March?
Solution:
There are 31 days in the month of March.

Example 4.
How many weeks are there in a year?
Solution:
There are 52 weeks in a year.

Example 5.
How many Fridays are there in the year 2021?
Solution:
There are 53 Fridays in the year 2021.

Reciprocal of a Fraction – Definition, Methods, Examples | How to find the Reciprocal of a Fraction?

Reciprocal of a Fraction

Reciprocal of a Fraction – The reciprocal of a fraction is nothing interchanging or switching the numerator and denominator. That means numerator becomes denominator and denominator becomes the numerator. In the case of a mixed fraction, you have to convert the mixed fraction to the improper fraction and then switch the numerator and denominator (top number to the bottom number). Learn how to find the reciprocal of a fraction with the help of the below examples.

Example: Suppose the fraction is a/b then the reciprocal of the fraction is b/a. Here b becomes numerator and a becomes denominator.

Do Refer:

How to find the Reciprocal of a Fraction?

Go through the simple process listed below to determine the reciprocal of a fraction. They are as follows

  • Initially, determine the numerator and denominator of a given fraction.
  • Fractions Reciprocal can be obtained by swapping or interchanging the numerator and denominators.
  • In the case of Mixed Fraction, you first need to change to improper fractions and then interchange the numerator and denominator of the improper fraction.

Reciprocal of a Fraction Examples

Example 1.
What is the opposite reciprocal of \(\frac{5}{6}\)?
Solution:
The opposite reciprocal of the fraction is nothing but changing the sign of the number. A positive number becomes a negative number.
So, the opposite reciprocal of the fraction \(\frac{5}{6}\) is –\(\frac{6}{5}\)

Example 2.
Find the reciprocal of the fraction \(\frac{2}{1}\)
Solution:
Given the fraction \(\frac{2}{1}\)
The reciprocal of a fraction is nothing interchanging or switching the numerator and denominator.
Thus the reciprocal of the fraction \(\frac{2}{1}\) is \(\frac{1}{2}\)

Example 3.
Find the reciprocal of the fraction \(\frac{17}{58}\)
Solution:
Given the fraction \(\frac{17}{58}\)
The reciprocal of a fraction is nothing interchanging or switching the numerator and denominator.
Thus the reciprocal of the fraction \(\frac{17}{58}\) is \(\frac{58}{17}\)

Example 4.
Find the reciprocal of the fraction \(\frac{16}{64}\)
Solution:
Given the fraction \(\frac{16}{64}\)
The reciprocal of a fraction is nothing interchanging or switching the numerator and denominator.
Thus the reciprocal of the fraction \(\frac{16}{64}\) is \(\frac{64}{16}\) or \(\frac{4}{1}\)

Example 5.
Find the reciprocal of the fraction \(\frac{2}{3}\)
Solution:
Given the fraction \(\frac{2}{3}\)
The reciprocal of a fraction is nothing interchanging or switching the numerator and denominator.
Thus the reciprocal of the fraction \(\frac{2}{3}\) is \(\frac{3}{2}\)

Example 6.
Find the negative reciprocal of the fraction \(\frac{7}{129}\)
Solution:
Given the fraction \(\frac{7}{129}\)
The opposite reciprocal of the fraction is nothing but changing the sign of the number. A positive number becomes a negative number.
Therefore the negative reciprocal of the fraction \(\frac{7}{129}\) is –\(\frac{129}{7}\)

Example 7.
Find the negative reciprocal of the fraction \(\frac{3}{5}\)
Solution:
Given the fraction \(\frac{3}{5}\)
The opposite reciprocal of the fraction is nothing but changing the sign of the number. A positive number becomes a negative number.
Therefore the negative reciprocal of the fraction \(\frac{3}{5}\) is –\(\frac{5}{3}\)

Example 8.
Write the reciprocal of the fraction \(\frac{3}{9}\)
Solution:
Given the fraction \(\frac{3}{9}\)
The reciprocal of a fraction is nothing interchanging or switching the numerator and denominator.
Therefore the reciprocal of the fraction \(\frac{3}{9}\) is \(\frac{9}{3}\) or \(\frac{3}{1}\) or 3.

FAQs on Reciprocal of a Fraction

1. What is a reciprocal of the fraction?

The reciprocal of a fraction will be obtained by interchanging the numerator and denominator.

2. Is 1 the reciprocal of 1?

Yes, 1 is the reciprocal of 1 itself. Since 1 can be written as \(\frac{1}{1}\)

3. What is the reciprocal of \(\frac{1}{3}\) as a fraction?

The reciprocal of the given fraction is \(\frac{3}{1}\) which means 3.

Estimating Sums and Differences – Definition, Examples | How to Estimate the Sum and Difference?

Estimating Sums and Differences

For Estimating Sums and Differences we use the concept of Rounding Off Numbers. Estimation is nothing but taking the values that is closer to the exact answer. Estimating Sums and Differences means writing answers that are approximately equal to the exact answer. Estimating the Values helps your child to improve mental math. Refer to the Solved Examples on Estimating the Sums and Differences explained step by step in the later modules.

Do Read:

How to Estimate the Sums and Differences of Whole Numbers?

Estimation means finding the answer closer to the accurate solution. The concept which is used for estimating addition and subtraction is round-off numbers. We can round the number nearest to ten, hundred, thousand, etc to estimate the answer. Bullet points to keep in mind is

  • If the number is less than 5, round down (means 0)
  • If the number is greater than 5, round up (means 1)

Advantages of Estimating Sum and Difference

There are many benefits of estimating sums and differences. Some of the advantages are shown below.

  • Estimating Addition and subtraction helps to improve mental math.
  • Your fluency in calculation will be improved.
  • You can understand the concept of rounding off numbers in the number system by learning the concept of estimation.

Estimating Sums and Differences Examples

Example 1.
Estimate the sum of 79, 89, 58.
Solution: 
9 is greater than 5, so you can add 1 to the tens place value and 0 to the unit place value.
The number 79 nearest to ten is 80
The number 89 nearest to ten is 90
The number 58 nearest to ten is 60
Now add three numbers 80 + 90 + 60 = 230
Now check whether the estimated answer is closer to the actual answer.
79 + 89 + 58 = 226
6 is greater than 5, so you can add 1 to the tens place value and 0 to the unit place value.
226 nearest to 10 is 230.

Example 2.
Estimate the difference between 219 and 17.
Solution:
9 is greater than 5, so you can add 1 to the tens place value and 0 to the unit place value.
7 is greater than 5, so you can add 1 to the tens place value and 0 to the unit place value.
219 nearest to ten is 220.
17 nearest to ten is 20.
Estimated difference is 220 – 20 = 200
Now check whether the estimated answer is closer to the actual answer.
219 – 17 = 202
202 nearest to ten is 200.

Example 3.
Estimate the sum and difference of 311 and 92.
Solution:
1 is less than 5, so you can round down to 0 to the unit place value.
2 is less than 5, so you can round down to 0 to the unit place value.
311 nearest to ten is 310
92 nearest to ten is 90
Estimated Sum: 310 + 90 = 400
Estimated Difference: 310 – 90 = 220
Now check if the estimated answer is closer to the actual answer.
311 + 92 = 403
403 is closer to 400.
311 – 92 = 219
219 is closer to 220.
So, the solution is correct.

Example 4.
Estimate the following additions and subtractions to the nearest ten, hundred and thousand.
i. 27 – 19
ii. 126 + 112
iii. 1002 + 996
iv. 2009 – 122
v. 39 – 12
Solution:
i. 27 – 19
9 is greater than 5, so you can add 1 to the tens place value and 0 to the unit place value.
7 is greater than 5, so you can add 1 to the tens place value and 0 to the unit place value.
27 to the nearest ten is 30.
19 to the nearest ten is 20.
Estimated Difference:
30 – 20 = 10
27 – 19 = 8
8 is closer to 10.
ii. 126 + 112
If the tens place is greater than 50, round up to the next hundred.
If the tens place is less than 50, round up to the previous hundred.
126 rounded to the nearest hundred is 100
112 rounded to the nearest hundred is 100
100 + 100 = 200
126 + 112 = 236
236 is closer to 200.
iii. 1002 + 996
If that digit is less than 5, you will round down to the previous thousand.
If that digit is greater than 5, you will round up to the next digit.
1002 to the nearest thousand is 1000.
996 to the nearest thousand is 1000.
1000 + 1000 = 2000
1002 + 996 = 1998
1998 is closer to 2000.
iv. 2009 – 122
If that digit is less than 5, you will round down to the previous thousand.
If that digit is less than 5, you will round down to the previous hundred.
2009 to the nearest thousand is 2000.
122 to the nearest hundred is 100.
2000 – 100 = 1900
2009 – 122 = 1880
1880 is closer to 1900.
v. 39 – 12
9 is greater than 5, so you can add 1 to the tens place value and 0 to the unit place value.
2 is less than 5, so you round down to 0 to the unit place value.
39 rounded to nearest ten is 40.
12 rounded to the nearest ten is 10.
40 – 10 = 30
39 – 12 = 27
27 is closer to 30.

Example 5.
Estimate the sum 711 and 625 to the nearest hundred.
Solution:
If that digit is less than 5, you will round down to the previous hundred.
If that digit is greater than 5, you will round up to the next hundred.
The unit place value is less than 5 so you have to round down to 0
711 number nearest to the hundred is 700.
The unit place value is equal to 5 so you have to round down to 0
625 number nearest to the hundred is 600.
700 + 600 = 1300
Now check if the estimated answer is closer to the actual answer.
711 + 625 = 1336
1336 is closer to 1300.

FAQs on Estimation of Addition and Subtraction

1. What is the actual difference and estimated difference?

If the exact difference is obtained, then it is called the actual difference. The estimated difference means the difference is obtained from the rounding off the given numbers.

2. How do you estimate the sum?

We estimate the addition by rounding off to the nearest numbers.

3. How do you estimate the difference?

We estimate the subtraction by rounding off to the nearest place values.

Indian Numbering System – Definition, Place Value Chart, Examples | Difference Between Indian and International Numeral Systems

Indian Numbering System

The Indian Numbering System is used in India to express large numbers. The terms like hundreds, thousands, lakhs, crores are the most commonly used terms to express the large numbers in Indian English. The zeroth power of 10 is 1, 10 power 1 is 10, 10 power of 2 is 100, the next powers of ten are called thousand, ten thousand, lakh, ten lakhs, crore.

Whereas in the Western System the next powers of ten are called one hundred thousand, one million, ten million, one hundred million, and so on. Get the solutions to the problems to calculate as per the Indian System or Indian Standard Number System.

Also, refer:

What is Indian Numbering System?

The Indian Numeral System is also known as Hindu-Arabic Numeral System. Indian Number System is a mathematical notation for expressing numbers, symbols, and digits or place values. In Arabic Number System or Hindu Number System ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 are used to write the numbers called digits. The Indian Numbering System groups the rightmost three digits together until hundreds place and then groups them by sets of two digits.

Indian Numbering System Chart

The Number System is a way to express or represent numbers. There are different place values like one’s place, tens place, hundreds place, thousands place, ten thousand places, lakhs place, and so on. In the Indian Number System separators or commas are used to identify the place values in the numbers. Know the way to express the large numbers in detail from this page.

Period Place Values Unit Digits Number of Digits
Tens Ones 1 1
Tens 10 2
Hundreds 100 3
Thousands Thousands 1000 4
Ten Thousands 10000 5
Lakhs Lakhs 100000 6
Ten Lakhs 1000000 7
Crores Crores 10000000 8
Ten Crores 100000000 9

Indian Number System Vs International Number System

The main difference between the Indian and International numeral systems is the placement of the separator. In the Indian System of Numeration, lakhs are written after thousands whereas in the International Numeral System millions are written after thousands.

100 thousand = 1 lakh
1 million = 10 lakhs
10 millions = 1 crore
100 millions = 10 crores

Placement of Commas in Indian Number System

In the Indian Number System, we use commas or separators to mark different periods. Commas help using reading and writing large numbers. According to the Indian Numbering System, the first comma is placed after the hundreds place, the second comma is placed after thousands period, the third comma is placed after the lakhs period, continue to place the commas after every two digits.

Example: Using Hindu-Arabic Numeral System read the number 1053671.
Solution: We can read the number 1053671 by placing the commas 10,53,671.
The word form is Ten Lakhs Fifty Three Thousand Six Hundred and Seventy One.

Indian Number System in Words | How to Write Number Names in Indian Number System?

Indian Number System in Words helps to read large numbers easily. Read the values according to the place values and commas. Neglect zero while read the number in word form. Go through the examples given below to get an idea about how to Write Number Names in Hindu-Arabic Number System.

Example: 999999
Solution:
First, we have to separate the numbers by using commas.
9,99,999
Word sentence for 9,99,999 is Nine Laks Ninety Nine Thousand Nine Hundred and Ninety Nine.

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Indian System of Numeration Examples

Example 1.
Write the place values for 10987654?
Solution:
In the number 10987654, the place value of each digit is
4 – units place
5 – tens place
6 – hundreds place
7 – thousands place
8 – ten thousands place
9 – lakhs place
0 – ten lakhs place
1 – crore place
1,09,87,654

Example 2.
Rewrite the number 18659 in Indian Numeral System?
Solution:
Use separators or commas to write the given number in the Indian Numeral System
18659 – 18,659

Example 3.
How do you write 6 lakhs 50 thousand in the Indian Number System?
Solution:
The method to writing 6 lakhs 50 thousand in Indian Number System is 6, 50, 000

Example 4.
How to write 100 hundred in Indian Numeral System?
Solution:
The method to write 100 hundred in Indian Numeral System 1,00, 000 (1 lakh)

Example 5.
How do you write 10 million in Indian Numeral System?
Solution:
10 Million can be written as 1,00,00,000

FAQs on Indian Numbering System

1. How do you write numbers in the Indian and International System?

In the international numbering system, millions are written after thousands while in the Indian system, lakhs are written after thousands

2. What do you mean by the Indian Number System?

The Indian numbering system corresponds to the Western system for the zeroth through the fourth powers of ten.

3. What is the difference between the Indian and International Numeral System?

The main difference between the Indian and International numeral system is the placement of commas

Uses of Roman Numerals | How and When to Use Roman Numerals? | Why are Roman Numbers Important?

Uses of Roman Numerals

Roman Numerals are the collection of Roman symbols that are used by the ancient Romans in the number system. Roman numerals are used for ranking as I, II, III, IV, and so on. These Roman Numerals are used to write classroom numbers. Also, it is used for writing the numbers on the clock. Know about various applications of Roman Numerals and why they are important, etc. in the further modules.

Do Refer:

Roman Numerals – Definition

Roman Numerals is the mathematical notation that does not follow the place value system. These Roman Numerals are used instead of Natural Numbers. C, D, I, L, M, V, X are the Roman Symbols that are used to express Roman Numerals. For better understanding, we have given the Roman Numerals Chart below.

Roman Symbol and its Equivalent Number

Uses of Roman Numerals in Everyday Life

  • The uses of Roman Numerals include year numbers on monuments and buildings, copyright dates on the title screens of movies and TV programs.
  • We also use Roman Numerals in writing the year numbers instead of Arabic Numbers.
  • Appendices or introduction of numbers is numbered with Roman Numerals.
  • The Roman Numerals are used to write the numbers in watches, clocks, etc.
  • Roman Numerals are used for writing classroom numbers like I, II, III, IV, V, VI, VII, VIII, IX, X.
  • Sequels of books are numbered with Roman Numerals.

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FAQs on Uses of Roman Numerals

1. Why are roman numerals important?

Thinking about numbers in different ways can also help them form connections or see patterns. Writing a number as a Roman numeral is another way to represent the numbers.

2. Where do we see Roman Numerals in our day to day life?

We can find Roman Numerals on Clocks, Watches, Books, Classroom numbers, etc.

3. When to Use Roman Numerals Instead of Numbers?

We use Roman Numerals instead of numbers in the introduction part or chapter numbers in books.