Category: Maths Notes PPT

1 is a very unique number. In classical theory it is neither considered even, odd or anything. It is just considered as a building block for any other number. 1 is so unique that it can not be included in any group nor in the group of prime numbers. It has been seen in the ancient time there was some ambiguity regarding inclusion of 1 in the set of prime numbers. Even the great mathematician G.H.Hardy seems to be in little confusion as he included 1 in the set of prime numbers in the first six editions of his book “A Course in Pure Mathematics” till 1933. But in 1938 he updated the inclusion and considered 2 to be the first prime number to start with.

Prime Number

Before knowing if 1 is a prime number or not, let us understand what is a prime number. A prime number may be an integer greater than 1 whose only factors are 1 and itself. An element may be an integer which will be divided evenly into another number. The few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 and so on. Numbers that have more than two factors are called composite numbers. The number 1 is neither prime nor composite.

For every prime p, there exists a major number p’ such p’ is bigger than p. This proof, which was demonstrated in the past by the Greek mathematician Euclid, validates the concept that there’s no “largest” prime. As the set of natural numbers N = {1, 2, 3, …} proceeds, prime numbers are generally subsided frequently and are harder to seek out in a reasonable amount of time. As of this writing, the most important known prime has 24,862,048 digits. It was discovered in 2018 by Patrick Laroche of the good Internet Mersenne Prime Search (GIMPS).

Properties of Prime Numbers

• Every number that’s greater than 1 is often divided by a minimum of one prime.

• Every even positive integer greater than the amount 2 is often expressed because of the sum of two primes.

List of Prime Numbers

Is 1 a Prime Number?

Number 1 has positive divisors for 1 and itself. consistent with the definition of a  prime number. Any number having only two positive divisors is referred to as prime numbers. So, is 1 a prime number or not? Is 1 a prime or composite number?

The answer to the present question is: No, 1 isn’t a major prime number and it’s not a composite number!

Lesson Summary:

Why is 1 not a Prime Number?

The answer to Why 1 is not a Prime Number is present in the definition for the prime numbers itself. For a number to be called the prime number, it must have only two of the positive factors. Now, for 1, the number of positive divisors or factors is only one that is 1 itself. So, this is why 1 is not a prime number here. But it is the most important number in Mathematics as it is the basic number used for forming other numbers.

 

Note: 2 is the smallest number that satisfies the definition for the prime numbers.

Solved Examples

Question 1: Which one of the following is a prime number?

Answer: 13 is a prime number because 13 has only two factors that are 13 and 1.

Question 2: Which one of the following is a prime number?

Answer: 3 is a prime number because 3 has only two factors that are 3 and 1.

Question 3: Which one of the following is a prime number?

Answer: 43 is a prime number because 43 has only two factors that are 43 and 1.

Question 4: Which one of the following is a prime number?

Answer: 53 is a prime number because 53 has only two factors that are 53 and 1.

You must have heard the term “probability” being coined for predicting the weather forecast in news TV bulletins for the next few days for some parts of the country. For calculating the probability of different types of situations, the probability formula and its related basic concepts are used. Probability is the way to measure the uncertainty, how likely an event has happened or is bound to happen.

Probability is that branch of mathematics that is concerned with the numerical description of how likely there are chances of the event to occur or how likely a particular proposition is true. For any event the probability lies between 0 to 1. 0 indicates the impossibility of the event to happen while 1 indicates certainty that the event is certain to occur.

Notation of Probability

For example: let us consider that two events are taking place namely A and B. So for the probability that event A can happen, we are going to write P(A) and for the probability that event B can happen, we can write P(B).

Terminologies Related to Probability Formula

There are a few crucial terminologies that are associated with all probability formulas

• Outcome: The result of an event after experimenting with the side of the coin after flipping, the number appearing on dice after rolling and a card is drawn out from a pack of well-shuffled cards, etc.

• Event: The combination of all possible outcomes of an experiment like getting head or tail on a tossed coin, getting an even or odd number on dice, etc.

• Sample Space: The set of all possible results or outcomes. This indicates that besides this there is no chance that any other result will come. Whatever the result is, it is from this sample Space only.

Probability Formulas with Examples

To Calculate the probability of an event to occur we use this probability formula, recalling, the probability is the likelihood of an event to happen. This formula is going to help you to get the probability of any particular event. 

This formula is the number of favourable outcomes to the total number of all the possible outcomes that we have already decided in the Sample Space.

The probability of an Event = (Number of favourable outcomes) / (Total number of possible outcomes)

P(A) = n(E) / n(S)

P(A) < 1

Here, P(A) means finding the probability of an event A, n(E) means the number of favourable outcomes of an event and n(S) means the set of all possible outcomes of an event.

If the probability of occurring an event is P(A) then the probability of not occurring an event is

P(A’) = 1- P(A)

Example 01: Probability of obtaining an odd number on rolling dice for once.

Solution: Sample Space = {1, 2, 3, 4, 5, 6}

n(S) = 6

Favourable outcomes = {1, 3, 5}

n(E) = 3

Using the probability formula,

P(A) = n(E) / n(S)

P(Getting an odd number) = 3 / 6 = ½ = 0.5

Important List of Probability Formulas

You just need to have the events for which you are looking for the probability and the formulas are going to make your work easier. In the formulas given below, we are taking 2 events namely A and B. The formulas are based on these events only.

P (A U B) = P (A) + P (B) – P (A ∩ B)

P (A ∩ B) = P (A) . P (B)

P(A NOT B) = A – B 

P(B NOT A) = B – A

Probability of occurrence of an event is P(A)

Probability of non-occurrence of the same event is P(A’). 

Some probability important formulas based on them are as follows:

• P(A.A’) = 0

• P(A.B) + P (A’.B’) = 1

• P(A’B) = P(B) – P(A.B)

• P(A.B’) = P(A) – P(A.B)

• P(A+B) = P(AB’) + P(A’B) + P(A.B)

Example 01: Two dice are rolled simultaneously. Calculate the probability of getting the sum of the numbers on the two dice is 6.

Solution: 

Sample Space= (1,1),(1,2),(1,3),(1,4),(1,5),(1,6)(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)(1,1),(1,2),(1,3),(1,4),(1,5),(1,6)(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)(6,1),(6,2),(6,3),(6,4),(6,5),(6,6){(1, 1), (1, 2), (1,3), (1,4), (1,5), (1, 6)} {(2, 1), (2, 2),(2,3), (2,4), (2,5), (2, 6)} {(3, 1), (3, 2), (3,3), (3,4), (3,5), (3, 6)} {(4, 1), (4, 2), (4,3), (4,4), (4,5), (4, 6)} {(5, 1), (5,2), (5,3), (5,4), (5,5), (5, 6)} {(6, 1), (6, 2), (6,3), (6,4), (6,5), (6, 6)} n(S) = 36 

Favourable outcomes = {(1, 5), (2, 4), (3, 3), (4, 2) and (5, 1)} 

n(E) = 5

Using, P(A) = n(E) / n(S)

P(Getting sum of numbers on two dice 6) = 5/ 36

The slope of a line calculates the “steepness” of a line. It is usually denoted by the letter m. So, the slope of a line is the change in Y divided by the change in X. As the change in Y is very high, the slope can range from zero to any number that we can think of. However, we usually have a maximum slope of positive or negative infinity. The change in x is much smaller than the change in y, which means that the change in x is much less than the change in y.

The slope of a line shows how slant the line is, how much the line rises vertically is compared with how much it runs horizontally. Being able to find the slope of a line, or using the slope to find points on the line, is an important skill used in economics, geoscience, accounting/finance, and other fields. The slope of a line is also defined as the ratio of rise over run. 

Uploaded soon)

The x and y coordinates of the lines are used to calculate the slope of the lines. It is the ratio of the change in the y-axis to the change in the x-axis.

The formula to calculate slope is given as,

That is, 

[frac{text{change in y Coordinates}}{text{change in x Coordinates}}]

Where m is the slope of the line. x1, x2 are the coordinates of x-axis and y1, y2 are the coordinates of y-axis.

The slope of the line can be a positive or negative value.

x and y are only used to identify the two points. They are not values or exponents, the points can be given any names.

[m=frac{y_{2}-y_{1}}{x_{2}-x_{1}}]

Slope formula when the general equation of a straight line is given

Uploaded soon)

If the general equation of a straight line is given as

ax + by + c  =  0,   

then, the formula for the slope of the line is 

m = – coefficient of x / coefficient of y 

m = -a/b

Slope Intercept Formula

Linear equations are “straight line” equations that have simple variable expressions with terms without exponents on them. You are dealing with a straight line equation, if you come across an equation with only x and y. To find the equation of a line and y-intercept in the steepness of the line, we use slope intercept formula.

Slope intercept formula is

The values used in formula are as follows:

Rules for Calculating the Slope of Line

Method 1:

Step 1: Find two points on the line.

Step 2: Count the rise. The number of units counts up or down to get from one point to the next. Record this number as your numerator.

Step 3: Count the run. The number of units counts left or right to get to the point. Record this number as your denominator.

Step 4: Simplify your fraction if possible.

Method 2:

To find out the slope of a line, we need only two points from that line, (x1, y1) and (x2, y2). 

There are three steps for calculating the slope of a straight line.

Step1: Identify two points on the line.

Step2: Select one to be (x1,y1) and the other to be(x2,y2).

Step3: Use the slope of the line formula to calculate the slope.

Some of the important points to remember to find the slope of the line. They are as follows:

• The slope formula can give a positive or negative result. 

• If the slope is a positive value, the line is in a rising state. 

• If the slope is a negative value, the line is descending. 

• Vertical lines have no slope.

• Horizontal lines have a zero slope.

• Parallel lines have equal slopes. 

• Perpendicular lines have negative reciprocal slopes. 

Solved Examples

Example 1: Find the slope of the line whose coordinates are (2,6) and (5,1).

Solution:

We have,

(x1, y1) = (2, 6)

(x2, y2) = (5, 1)

The slope of a line formula is [m=frac{y_{2}-y_{1}}{x_{2}-x_{1}}]

m = (1 − 6/ 5 − 2)

m = −5/3

m = − 1.666

Example 2: If the slope of a line passing through the points (4, x) and (2, -7) is 3, then what is the value of x?

Solution:

We have

Slope = m = 3

Points:

(x1, y1 ) = (4, x)

(x2, y2) = (2, -7)

We know that,

Slope [m=frac{y_{2}-y_{1}}{x_{2}-x_{1}}]

3 = (-7 – x)/(2 – 4)

3 = (-7 – x)/(-2)

or, -7 – x = 3(-2)

or, -7 – x = -6

so, x = -7 + 6 = -1

Therefore, the value of x = -1.

The F Test Formula is a Statistical Formula used to test the significance of differences between two groups of Data. It is often used in research studies to determine whether the difference in the means of two populations is Statistically significant.

It is based on the F Statistic, which is a measure of how much variation exists in one group of Data compared to another. Students who are studying for their Statistics course will need to be familiar with this Formula. Our article will provide a detailed explanation of how to use the F Test Formula. It will also provide examples of how to use it in practice.

The use of the F Test Formula is a critical step in any research study, and it is important to understand how to use it correctly. You will be able to find the F Test Formula in most Statistics textbooks.

What is the Definition of F-Test Statistic Formula?

It is a known fact that Statistics is a branch of Mathematics that deals with the collection, classification and representation of Data. The tests that use F – distribution are represented by a single word in Statistics called the F Test. F Test is usually used as a generalized Statement for comparing two variances. F Test Statistic Formula is used in various other tests such as regression analysis, the chow test and Scheffe test. F Tests can be conducted by using several technological aids. However, the manual calculation is a little complex and time-consuming. This article gives an in-detail description of the F Test Formula and its usage.

Definition of F-Test Formula

F Test is a test Statistic that has an F distribution under the null hypothesis. It is used in comparing the Statistical model with respect to the available Data set. The name for the test is given in honour of Sir. Ronald A Fisher by George W Snedecor. To perform an F Test using technology, the following aspects are to be taken care of.

• State the null hypothesis along with the alternative hypothesis.

• Compute the value of ‘F’ with the help of the standard Formula.

• Determine the value of the F Statistic. The ratio of the variance of the group of means to the mean of the within-group variances. 

• As the last step, support or reject the Null hypothesis.

F-Test Equation to Compare Two Variances:

In Statistics, the F-test Formula is used to compare two variances, say σ1 and σ2, by dividing them. As the variances are always positive, the result will also always be positive. Hence, the F Test equation used to compare two variances is given as:

F_value =[frac{variance1}{variance2}]

i.e. F_value = [frac{sigma_{1}^{2}}{sigma_{2}^{2}}]

F Test Formula helps us to compare the variances of two different sets of values. To use F distribution under the null hypothesis, it is important to determine the mean of the two given observations at first and then calculate the variance. 

[sigma ^{2}=frac{sum (x-bar{x})^{2}}{n-1}]

In the above formula, 

σ2 is the variance

x is the values given in a set of data

x is the mean of the given Data set 

n is the total number of values in the Data set

While running an F Test, it is very important to note that the population variances are equal. In more simple words, it is always assumed that the variances are equal to unity or 1. Therefore, the variances are always equal in the case of the null hypothesis.

F Test Statistic Formula Assumptions

F Test equation involves several assumptions. In order to use the F – test Formula, the population should be distributed normally. The samples considered for the test should be independent events. In addition to these, it is also important to consider the following points.

• Calculation of right-tailed tests is easier. To force the test into a right-tailed test, the larger variance is pushed in the numerator.

• In the case of two-tailed tests, alpha is divided by two prior to the determination of critical value. 

• Variances are the squares of the standard deviations.

If the obtained degree of freedom is not listed in the F table, it is always better to use a larger critical value to decrease the probability of type 1 errors. 

F-Value Definition: Example Problems

Example 1: 

Perform an F Test for the following samples.

1. Sample 1 with variance equal to 109.63 and sample size equal to 41.

2. Sample 2 with variance equal to 65.99 and sample size equal to 21.

Solution: 

Step 1:

The hypothesis Statements are written as:

H_0: No difference in variances 

H_a: Difference invariances 

Step 2: 

Calculate the value of F critical. In this case, the highest variance is taken as the numerator and the lowest variance in the denominator.

F_value = [frac{sigma_{1}^{2}}{sigma_{2}^{2}}]

F_value = [frac{109.63}{65.99}]

F_value = 1.66

Step 3:

The next step is the calculation of degrees of freedom.

The degrees of freedom is calculated as Sample size – 1

The degree of freedom for sample 1 is 41 -1 = 40.

The degree of freedom for sample 2 is 21 – 1 = 20.

Step 4:

There is no alpha level described in the question, and hence a standard alpha level of 0.05 is chosen. During the test, the alpha level should be reduced to half the initial value, and hence it becomes 0.025.

Step 5:

Using the F table, the critical F value is determined with alpha at 0.025. The critical value for (40, 20) at alpha equal to 0.025 is 2.287.

Step 6:

It is now the time for comparing the calculated value with the standard value in the table. Generally, the null hypothesis is rejected if the calculated value is greater than the table value. In this F value definition example, the calculated value is 1.66, and the table value is 2.287. 

It is clear from the values that 1.66 < 2.287. Hence, the null hypothesis cannot be rejected.

Fun Facts About F-Value Definition:

• In the case of Statistical calculations where the null hypothesis can be rejected, the F value can be less than 1; however, not exactly equal to zero.

• The F critical value cannot be exactly equal to zero. If the F value is exactly zero, it indicates that the mean of every sample is exactly the same, and the variance is zero. 

• One of the key points to remember while working with the F Statistic is that the population variances are always considered to be equal. If this condition is not met, the obtained F value might not be correct.

• The degrees of freedom is taken as the number of samples minus one. In the case of a two-sample problem, there are two samples, and hence it becomes 2 – 1 = 1.

• When the alpha level is not mentioned in the F Test, the standard value used in most of the cases is equal to 0.05.

Conclusion

In case of a problem with two sample Data sets, the F value can be obtained by dividing the larger variance by the smaller one. In order to perform a test at a pre-specified alpha level, it is always better to use standard values from the F table rather than using calculated values. The F value definition example has demonstrated how to calculate the F Statistic along with the relevant steps and interpretation of results. Students can use the F Statistic Formula to understand how it is used for t-test calculations. t-value definition examples are also available on this website. You can download the F table pdf to perform your own calculations.

In mathematics, we usually need to find the derivative of some mathematical functions. It gives the rate of change of one variable with respect to others. Integration is the opposite process of differentiation. The fundamental use of integration is to get back the function whose derivatives are known. So, it is like an antiderivative procedure. Thus, integrals can be computed by viewing an integration as an inverse operation to differentiation. In this article we are going to discuss the concept of integration, basic integration formulas, integration formula of uv,integration formula list  as well as some integration formula with examples. Let us learn it!

Concept of Integration

• In Mathematics, when general addition operations cannot be performed, we use integration to add values on a large scale.

• There are various different methods in mathematics that can be used to integrate functions.

•  Integration and differentiation are known to be a pair of inverse functions similar to  addition- subtraction, and multiplication-division in Mathematics. 

• The process of finding functions whose derivative is given is known as integration or anti-differentiation.

[Image to be added Soon]

Here’s What Integration is!

Concept of Integration

Integration is the algebraic method to find the integral for a function at any point on the graph. Finding the integral of some function with respect to some variable x means finding the area to the x-axis from the curve. Therefore, the integral is also known as the anti-derivative because integrating is the reverse process of differentiating.

Not only the integral comes to determine the inverse process of taking the derivative. But it comes in the picture for solving the area problem as well. Similar to the process of differentiation for finding the slope at any point on the graph, this process of integration will be used to find the area of the curve up to any point on the graph.

The integral of the function of x from range a to b will be the sum of the rectangles to the curve at each interval of change in x as the number of rectangles goes to infinity.

We can write the integral of a function f(x) with respect to x :

[int]f(x)dx

Also, integration is often considered as an inverse to the differentiation operation means that if,

[frac{d}{dx}] (f(x)) = g(x) 

Then,

[int]g(x)dx  = f(x) +C  

The extra  variable C is known as the constant of integration, which is really necessary. The constant of integration is important because that after all differentiation kills off constants, that is why integration and differentiation are not known as exactly inverse operations of each other.

Since the process of integration is known to be almost the inverse operation of differentiation, the recollection of formulas and processes for differentiation is often possible. So, many differentiation formulae can be used in Mathematics to provide the corresponding formula for the integration process.

Points to Remember 

• Since the integral of a function isn’t definite, therefore it can be referred to as indefinite integral.

• The integral of a function at a point can never be found; we always find the integral of a given function in an interval.

• The Integral of a function is not unique; that is integrals of a function differ by numbers.

Here’s a List of Integration Methods

1.Integration by Substitution

2. Integration by Parts

3.Integration by Partial Fraction

4.Integration of Some particular fraction

5.Integration Using Trigonometric Identities

1. Integration by Substitution

            I = f(x).dx =f(g(t).g'(t).dt

2. Integration by Parts

The integration formula of uv : ∫f(x).g(x).dx = f(x).∫g(x).dx–∫(f′(x).∫g(x).dx).dx

Which can be further written as integral of the product of any two functions =  (First function × Integral of the second function) – Integral of [ (differentiation of the first function) × Integral of the second function]. This is the integration of the uv formula.

3. Integration of Some Particular Function

1. ∫ dx/ (x2 – a2) is equal to 12 a log | (x – a) / (x + a) | + c

2. ∫ dx/ (a2 – x2) is equal to1 2 a log | (a + x) / (a – x) | + c

3. ∫ dx / (x2 + a2) is equal to1a tan–1 (x/a) + c

4. ∫ dx /√ (x2 – a2) is equal to log| x+√(x2 – a2) | + c

5. ∫ dx /√ (a2 – x2) is equal to sin–1 (xa) + c

6. ∫ dx /√ (x2 + a2) is equal to  log | x + √(x2 + a2) | + c

Definite Integrals

Definite integrals are known as the special kind of integration, where both endpoints are fixed( that is both a and b are given). So, it always represents some bounded region, for computation. 

Definite Integral Formula

[int_{a}^{b}]f(x)dx= F(b)-F(a)   

Standard Integrals in Integration( Basic Integration formulas)  –

Integration Formula List

Questions to be Solved

Q1: Solve the following definite integral.

                    [int_{-2}^{3}] x³ dx

Solution:– [int_{-2}^{3}] x³ dx

[int_{-2}^{3}] x3 dx = [[frac{x^{4}}{4}]][_{-2}^{3}]

= [frac{81}{4}] – [frac{16}{4}]

= [frac{65}{4}]

= 16.25

An equation of the form ax2 + bx + c = 0, where a, b, c are real numbers and a0 is called a quadratic equation. The value of unknown variable x, which satisfies the given quadratic equation is called the roots of quadratic equation. For example, if is a root of quadratic equation ax2 + bx + c = 0, then a2 + b + c = 0. And the process of finding roots is known as solving a quadratic equation. There are three methods to solve a quadratic equation, which are as follows:

1. Solving a quadratic equation by factorisation.

2. Solving a quadratic equation by completing the square.

3. Solving a quadratic equation using quadratic formula.

In this article, we will learn about quadratic formula, its derivation and how to solve a quadratic equation using quadratic formula. 

What is Quadratic Formula?

Quadratic formula is a formula that helps us to find the roots of a quadratic equation very easily by replacing the other methods of finding the roots like, factorisation method, completing the square method. The quadratic formula to find the roots of quadratic equation ax2 + bx + c = 0, where a ≠ 0 is given by:

[x = frac{-b pm sqrt{b^{2} – 4ac}}{2a}]

The plus (+) and minus (-) sign represents that quadratic formula will give two roots, one root corresponding to the plus (+) sign and another root corresponding to the minus (-) sign.

i.e.,  [x_{1} = frac{-b – sqrt{b^{2} – 4ac}}{2a}] and [x_{2} = frac{-b + sqrt{b^{2} – 4ac}}{2a}]. Both roots are evaluated by substituting the corresponding values of coefficients a, b and c from the quadratic equation ax2 + bx + c = 0.

Derivation of Quadratic Formula

Consider the quadratic equation ax2 + bx + c = 0, where a, b, c are real numbers and a0. Then,

ax2 + bx + c = 0

⇒ ax2 + bx = –c

On dividing both sides by a, we get:

⇒ [x^{2} + frac{b}{a}x = frac{-c}{a}]

Now adding [(frac{b}{2a})^{2}] on both sides, we get:

⇒ [x^{2} + frac{b}{a}x + (frac{b}{2a})^{2} = frac{-c}{a} + (frac{b}{2a})^{2}]

⇒ [(x  + frac{b}{2a})^{2} = (frac{-c}{a} + frac{b^{2}}{4a^{2}})]

⇒ [(x + frac{b}{2a})^{2} = frac{(b^{2} – 4ac)}{4a^{2}}]

⇒ [(x – frac{b}{2a}) = frac{pm sqrt{b^{2} – 4ac}}{2a}], when (b2 – 4ac) 0

⇒ [x = – frac{b}{2a} pm frac{sqrt{b^{2} – 4ac}}{2a}]

⇒ [x = frac{-b pm sqrt{b^{2} – 4ac}}{2a}]

Therefore, the roots of quadratic equation ax2 + bx + c = 0 is  [x_{1} = frac{-b + sqrt{b^{2} – 4ac}}{2a}] and [x_{2} = frac{-b – sqrt{b^{2} – 4ac}}{2a}].

Discriminant of Quadratic Equation

The expression D = (b2 – 4ac) is called the discriminant of quadratic equation ax2 + bx + c = 0. 

The roots of ax2 + bx + c = 0 are real only when D 0 i.e., b2 – 4ac 0 and the roots are given by [x_{1} = frac{-b + sqrt{D}}{2a}] and [x_{2} = frac{-b – sqrt{D}}{2a}].

How to Solve Quadratic Equations using Quadratic Formula?

To learn how to solve quadratic equations using quadratic formula, let us consider some examples and solve them using quadratic formula.

Example 1: Find the roots of quadratic equation 15x2 – x – 28 = 0 using quadratic formula.

Solution:

The given quadratic equation is 15x2 – x – 28 = 0. Comparing it with ax2 + bx + c = 0, we get a = 15, b = -1 and c = -28.

So, D = b2 – 4ac = (-1)2 – 4 × 15 × (-28) = 1681. As D = 1681 > 0, The given quadratic equation has real roots.

Now substituting the corresponding values of a, b and c in quadratic formula: [x = frac{-b pm sqrt{b^{2} – 4ac}}{2a}], we get,

[x = frac{-(-1) pm sqrt{(-1)^{2} – 4 times 15 times (-28)}}{2 times 15}]

⇒ [x = frac{1 pm sqrt{1681}}{30}]

For plus (+) sign, 

The root is [x_{1} = frac{1 + sqrt{1681}}{30} = frac{1 + 41}{30} = frac{42}{30} = frac{7}{5}].

and, for minus (-) sign, 

the root is [x_{2} = frac{1 – sqrt{1681}}{30} = frac{1 – 41}{30} = frac{-40}{30} = frac{-4}{3}]

Hence, the required roots of quadratic equation 15x2 – x – 28 = 0 are 7/5 and -4/3 .

Example 2: Find the roots of quadratic equation x2 + 6x + 6 = 0 using quadratic formula.

Solution:

The given quadratic equation is x2 + 6x + 6 = 0. Comparing it with ax2 + bx + c = 0, we get a = 1, b = 6 and c = 6.

So, D = b2 – 4ac = (6)2 – 4 × 1 × 6 = 12. As D = 12 > 0, The given quadratic equation has real roots.

Now substituting the corresponding values of a, b and c in quadratic formula: [x = frac{-b pm sqrt{b^{2} – 4ac}}{2a}], we get,

[x = frac{-(-6) pm sqrt{(6)^{2} – 4 times 1 times 6}}{2 times 1}]

⇒ [x = frac{-6 pm sqrt{12}}{2}]

For plus (+) sign, 

the root is [x_{1} = frac{-6 + sqrt{12}}{2} = frac{-6 + 2sqrt{3}}{2} = (-3 + sqrt{3})]

and, for minus (-) sign, 

the root is the root is [x_{2} = frac{-6 – sqrt{12}}{2} = frac{-6 – 2sqrt{3}}{2} = (-3 – sqrt{3})] 

Hence, the required roots of quadratic equation x2 + 6x + 6 = 0 are [((-3 + sqrt{3})] and  [(-3 – sqrt{3})].