Solving the equations of the second degree

Solving the equations of the second degree is quite an important skill in mathematics. Many life situations can be modelled with the second degree equations

The objective of this unit is to show you some of the most important technique for factor equations of the second degree. See below:

Important note:

Before starting this unit, it is very important to study, review or what factoring is

Technique # 1:

Solve equations of the second degree by factoring
This technique shows you how to solve the equations of the form ax2 + bx + c = 0 when a = 1 or more

When a = 1, it’s quite simple!

When one is equal to 1, it is a bit tedious. You need to do the trials and errors

Technique # 2:

Address by completing the square
This technique shows you how to fill the place to solve second degree equations

Particular attention to all stages and to learn more!

The evidence even of the quadratic formula is entirely based on this technique.

Make sure that you understand, or you can fight a lot to understand the proof of the quadratic formula

Technique # 3:

Solve using the quadratic formula
Technique # 1 (a) of the limits in the resolution of the equations of the second degree. When the answers are not whole numbers, but the real numbers, it is very difficult or almost impossible to find solutions.

With this formula, you can solve the equations of the second degree and it does not matter how complex the equation is or how weird is the answer.

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Purchase officer position description

Post purchase officer and introduction to the field description:

An agent to purchase, you will work in a fast-pace environment characterized by high energy.

Purchasing agents buy things for businesses, such as the contents first, products and other services that the company must maintain operations.

For example, in a school they may have to buy paper towels, books and school supplies, ordered by teachers, food, desks and so on…

whatever they ask, their job is to ensure that the organization works very well in ordering stuff in a timely manner.

The challenge:

Although the mathematics in a valuable asset in this trade, it is not limited to be good in math.

The work of buyers is more complex than the simple purchase. Because the market is always seeking and compete for buyers, decisions are often made quickly; This can create financial risks.

An organization expected the officer not only know how to buy, but they have to take important decisions on the quality of the articles that they buy.

Buy it cheap, and it cannot be of good quality. Buy too high, and there the company may lose money.

Yes, these people are an essential element of the management team and their contribution is essential, because they make a decision which may affect the financial well-being of the organization.

Your mathematical skills will become very handy when you need to calculate the cost of an item and what will be the cost for the transport of this element.

Once more, the company must more that your mathematical skills because it is your responsibility to ensure that vendors will be will meet their promises on the quantity and delivery date. It can help your organization a lot to get what they need in a timely manner.

That said, the agent must anticipate problems-bad weather, transportation-related problems and any other which may defer or delay delivery.

Get the job:

Mathematical modeling is used by almost all business strategists to anticipate the results to changes in variables.

The varables could be the price of raw materials, cost of advertising, packaging and shipping costs, and many other variables which may affect the success of a company or organization.

Have a Bachelor’s degree in mathematics will fit very well into the needs of these organizations working under these pressures.

Many employers are desperately looking for people with mathematical skills. You may not be aware of the demand placed on you. You will do well if you believe that your potential.

Keep in mind that a master is required for numerous upper level manager positions.

If you enter the profession, don’t be surprised if they ask you to show your certification because it has become increasingly important.

For more information about the purchase of the certification and the agent job description, you can consult the following Web sites:

Institute of supply management

American purchasing society

The association for operations management

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Professor of mathematics at lycée

Introduction to the field:

Before becoming a Professor of mathematics at lycée, licensing and certification are required in all fifty States.

States most wants to a concentration of zone object and a major in education for teaching in high school.

Some schools balancing courses in mathematics and education. In some other schools, you will need to complete all the courses of a full major in mathematics.

All certified graduate teachers are qualified for jobs in education entry level. Be inexperienced can be much easier to get a job.

Because of tight budgets, the principal will be drawn by someone who has no experience and is ready to accept a lower salary to start.

After obtained employment, continue you normally to take training courses continue to be eligible to renew their licence and keep their job.

Failure to do so may result in loss of your work at the discretion of the School Board.

Check the Ministry of education. Then, locate the DOE in your State and discover the specific to renew your license.

You can be a Professor of mathematics at lycée certified even if you do not have a master’s degree in education. Many States give you a test. The test is usually called the Praxis series

If you pass the test, you will be eligible to receive a preliminary or provisional licence. Usually, it is valid for three to five years according to the State.

During this time, they expect you to take education classes to obtain your master.

If you work full time, do not panic! Often, training can be obtained in distance learning classes in the comfort of your home. In addition, several school districts will pay for your studies. So find the right one.

A question which often comes is: “is it possible to start teaching without certification from the State and with a Bachelor’s degree in mathematics.”

The answer is Yes, in certain circumstances. When States have difficulties in maintaining their teachers because of the location and/or the pay scale, they can give you a waiver. The waiver may be renewed with satisfactory progress.

Satisfactory progress means that Edem show you DOE and the school district you will become certified soon, usually in a year or two.

Working condition can vary considerably.A Professor of mathematics at lycée teaches time full typical perhaps the up to five or six are classes for a day.

The teacher may also have supervision room dining or the room of study during the week. Teachers may also be responsible for study club, after school club competition of mathematics, detention centres or even a sporting activity monitor according to the contract or school.

The most difficult element, a Professor of mathematics at lycée may have to face is that of the discipline in the classroom. You were a student yourself. You should know that attendance is not voluntary, student show resistance and acting is not uncommon.

If you want to be an effective teacher, it is imperative that you master classroom management.

Secondary math teacher day is full. You will have a schedule that say you that you will start at the end. However, you will have one or two planing time per day.

During this time, teachers are supposed to lesson for the next plan, keeping grades, attendance and progress in due form.

When school is over, your work day is no more. Occasionally, teachers may accompany students on the field trip.

When they finally return home, they may have some planing more to do for the next day.If you want to be a College Professor instead, see Professor for more information.

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Distance formula

Given two points (x 1, y1), (x 2, y2) the formula for distance is calculated with the following formula.


Example # 1:

Use the formula distance to find the distance between (2,3) and (6.6)

Let (x 1, y1) = (2,3)

Let (x 2, y2) = (6.6)

Example # 2:

Use the remote to find the distance between (17,12) and form (9.6)

Let (x 1, y1) = (17,12)

Let (x 2, y2) = (9.6)

Example # 3:

Use the remote to find the distance between (17,12) and form (9.6)

Let (x 1, y1) = (7.8)

Let (x 2, y2) = (4.5)

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What is a computer programmer? Discover here.

Introduction to the field:

What is a computer programmer? A computer programmer to write code or instructions to perform specific tasks on the computer.

These instructions are called programs. Before writing the code, the programmer has obtain algorithms for a software engineer

To write these instructions, programmers using languages such as Java, C, C++ or COBOL programming.Today, Java is used more and more.

For example, instructions can be granted to the computer that the computer to solve linear equations or the second degree equations.

Our example here is simplified. Programming involves much more complicated tasks.

To program a scientific calculator, for example, may require a solid understanding of a programming language and mathematics at the same time. Very often, what is behind a complicated code is a sophisticated mathematical concept.

The challenge:

Programmers should have the ability to think logically, to use analytical skills and attention to detail.

The work also requires a person to be patient and persistent, because the computer instructions do not work only all the time. Some ingenuity may be sometimes necessary to make things work.

Programmers spend most of their time behind a computer. Long hours of work or weekend work may be required to resolve unexpected problems or to complete a long project.

As a result, programmers are susceptible to eyestrain, back issues, or wrist problems such as carpal tunnel syndrome.

In addition, the programmer must continually further their skills and their knowledge because of the programming industry changing.We have said before that Java is widely used in programming.

Don’t be surprised if an another programming language, more powerful than Java, is designed in the coming decades or even earlier.

Get the job:

It is not impossible to get a job with a degree associate for some positions. However, a Bachelor’s degree is usually required for most jobs programs.

If you have your Bachelor’s degree in mathematics, you will be very attractive for employers. Sooner or later, it is important to become certified.

Certification has emerged to make sure that you are sufficiently competent to carry out your work.

For more information about certification, visit this Web site which explains how to become a certified computer programmer

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Professor of mathematics

Introduction to the field:

Usually, a Professor of mathematics, also known under the name of post-secondary level of teacher, teaches at a community college or University.

The work includes plans lesson, giving lectures to students approximately 12 to 16 hours of class per week and parameter about 3 to 7 hours per day to meet to respond to questions and concerns.

The environment is completely different from that of high school and middle school.

Most of the students who go to College are aware of the importance of education.

Therefore, classroom behavior, such as speaking, acting, disturb the other students is unlikely.

As a result, postsecondary teachers find satisfaction and pleasure in teaching.

In addition, classes, Office hours, research and meeting are part of their daily schedule and their schedule is not pre-determined as high school.

The challenge:

The main concern of every teacher post-secondary is obtaining tenure. Tenure is not given to each teacher at the post-secondary level in the profession.

The granting of tenure is established after a post-secondary education has documented significant teaching stories, campus Committee work, publications, and obtained excellent student reviews.

The whole process can take approximately 7 years. At the end of the trial period, if the teacher did not mandate, the teacher may be asked to leave the institution.

Another thing to keep in mind is that College and University hire new tenure-track faculty as instructors or assistant professors. The title of Associate Professor or Professor of mathematics is granted with time and experience.

Finally, postsecondary teachers must conduct scholarly research in the discipline they teach. Publications are reviewed on a regular basis by the Department Director and academic Deans.

This is important because this can help determine whether a teacher should be granted tenure or promotion in their careers.

Get the job:

To obtain post-secondary as teacher employment, a doctoral degree is that many employers are looking for.

At a Community College, someone can get a position teaching either with a master’s degree in mathematics or mathematics education.

However, a master’s degree in mathematics is more desirable than in the teaching of mathematics.

When the need is urgent and pressing, some community colleges hire folks with only a Bachelor’s degree in mathematics, provided that the teacher has a lot of experience.

One day, I was browsing the Web site of Massasoit Community College and I saw an opening in mathematics education with only a Bachelor’s degree in mathematics

The position was temporary (one year). However, it shows something interesting!

When there is a great need, you can get your foot in the door, as a Professor of mathematics at Community College with just a bachelors

Keep in mind that, even if take you with a Bachelor’s degree, you will be required to obtain your master’s degree in mathematics in a period of time

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Evidence of recurrence

To evidence by recurrence, follow the steps below exactly as shown, in the given order:

Step # 1:

Show this is true for n = 1, n = 2…

Step # 2:

Suppose that this is true for n = k

Step # 3:

To prove this is true for n = k + 1

Notes the explanations on evidence by recurrence and important:

Step # 1, you try to show that this is true for specific values. You are free to do this test with a single value or fifty values of your choice or more.

However, showing this is true for the values of one million or more does not prove it will be true for all values. It is a very important observation!

In step # 2, since you have already shown that this is true for one or more values, it is logical to assume or assumes that this is true for n = k or generally.

We usually use the asumption that here we complement or prove step # 3

Step # 3, finally show you that it is true for all values. Note that this step # 2 does not show that this is true for all values.


Show that for any n, 2 + 4 + 6 +… + 2n = n (n + 1)

Step # 1:

Display the equation is true for n = 1, n = 2…

There is a pitfall to avoid here.

n = 1 means the first value of the expression on the left side. In this case 2

n = 2 means the first two values of the expression on the left side. In this case 2 + 4.

n = 3 means the first three values in the expression on the left side. In this case 2 + 4 + 6

Thus, showing the equation 2 + 4 + 6 +… + 2n = n (n + 1) is true for n = 4 means we need to show that 2 + 4 + 6 + 8 = 4 (4 + 1)

2 + 4 + 6 + 8 = 6 + 6 + 8 = 12 + 8 = 20 and 4 (4 + 1) = 4 × 5 = 20

Since the left side is equal to the right (20 = 20), step # 1 is done. It is not necessary to choose other values that you could do it just for fun and to prove to yourself that it will work for other values.

Step # 2:

Assume that the equation is true for n = k

Just replace n by k.

2 + 4 + 6 +… + 2 k = k (k + 1)

Step # 3:

To prove the equation is true for n = k + 1

It is the most difficult part of the evidence by recurrence. Things can get really hard here. Step in this well problem!

At this stage, you need to write what it means for the equation be true for n = k + 1

Be careful! Just because you wrote what means does not mean that you have proved it. There is another pitfall to avoid when you work on a proof by recurrence.

What this means:

After replacing k by k + 1, you get:

2 + 4 + 6 +… + 2 × (k + 1) = k + 1 (k + 1 + 1)

2 + 4 + 6 +… + 2 × (k + 1) = k + 1 (k + 2)

2 + 4 + 6 + … + 2 × ( k + 1) = ( k + 1 ) × ( k + 2)

We will give you a summary because you may have lost tract of what we are trying to do here.

We do not prove anything yet. The equation 2 + 4 + 6 +… + 2 × (k + 1) = (k + 1) × (k + 2) is what it means for the equation be true for n = k + 1

We are now ready to complete the proof by recurrence using the hypothesis in step no. 2.

on the assumption, 2 + 4 + 6 +… + 2 k = k (k + 1)

Say to yourself, “that the next term is seeking as”?

Since the latter term is now 2 k, the next term should be 2 × (k + 1)

Add 2 × (k + 1) on both sides of the hypothesis

2 + 4 + 6 +… + 2 k + 2 × (k + 1) = k (k + 1) + 2 × (k + 1)

= k2 + k + 2 k + 2.

= k2 + 3 k + 2.

Since 2 = 1 × 2 and 1 + 2 = 3,

k2 + 3k + 2 = ( k + 1) × ( k + 2)

Therefore, 2 + 4 + 6 +… + 2 k + 2 × (k + 1) = (k + 1) × (k + 2) and evidence of recurrence is complete!

The above is a solid and well explained by recurrence. Although study!

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