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The coefficients of a, b, c and d are real or complex numbers with a not equals to zero (a ≠ 0). It must have the term x3 in it, or else it will not be a cubic equation. But any or all of b, c and d can be zero.
Mathematics 9 Lesson 1-D: System of Equations Involving Quadratic EquationsJuan Miguel Palero
This powerpoint presentation discusses or talks about the topic or lesson System of Equations involving Quadratic Equations. It also discusses and explains the rules, steps and examples of System of Equations involving Quadratic Equations
Mathematics 9 Lesson 1-D: System of Equations Involving Quadratic EquationsJuan Miguel Palero
This powerpoint presentation discusses or talks about the topic or lesson System of Equations involving Quadratic Equations. It also discusses and explains the rules, steps and examples of System of Equations involving Quadratic Equations
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Interesting comparative essay topics, tips and suggestionslevibaxter
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From understanding the type of comparison you are supposed to make to jotting down the vital aspects of the topic, composing comparative essays take a lot of effort, concentration and a clear understanding of facts and figures before one decides to write the entire paper. If you are currently stuck on a comparative essay topic and finding things difficult to organize and add in your paper, then take a look at the following suggestions and helpful tips that can assist you to come up with convincing and well-framed comparative essay assignments.
A handbook on how to write a conclusion for an essaylevibaxter
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“How to write a conclusion for an essay!” – Well, the above outline must have helped you in getting over these worries. Here are some tips on the same context. You must apply the following principles in your essay while writing the conclusion section.
Do not simply summarize your essay, keep it brief and effective
Remember to clearly rephrase and restate your argument in the ending section
Draft the ending depending on your discipline
You should not quote filler texts rather choose to be relevant
Keep a formal tone as there is no need to be too emotional
Engaging essays examples and a complete how to guidelevibaxter
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Writing an essay perfectly is no mean feat. There has to be a considerable amount of preparation to bring out the best of the writing acumen in yourself. Some of the ways to do it is taking a great stance after choosing the topic and conducting background study thoroughly. Here is how you can prepare for writing an essay that leaves a lasting impression.
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You must have heard about proofreading and editing several times, but do you really know whether they are the same or different? If you also believe these two tasks are the same, then, you need to change that thought immediately. Yes! Both are completely different. There are significant differences between proofreading and editing. To develop a refined and superior-quality paper, both of these processes are very important. If you are still confused about the two, then, read this blog.
Tips on writing an essay in a short spanlevibaxter
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How to write an essay fast? Adopting the right mindset is one of the vital steps you must follow. It is obvious that you are feeling nervous and anxious in the wee hours before submission. Unless you banish all these negative thoughts, you cannot work methodically and logically. Breathe in, relax, and relish the challenge in the most positive way you can.
Nature vs. nurture a rundown of the never ending debatelevibaxter
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According to French philosopher, Rene Descartes, “We all, as individual human beings, have certain innate ideas that enduringly underpin our approach to the world.” And if we are to believe that, then what exactly influences our innate ideas? Is it heredity, or is it the environment we grow up in?
8 tips to prepare essays for scholarship college funding pptlevibaxter
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Have you always harbored the dream of studying in an Ivy League college, even if you don’t have sufficient resources? Do you feel that your career goals will be restricted because you can’t afford the hefty expenses of the college tuition fees? Well, fortunately for you, various renowned colleges and universities in the USA are offering scholarship support and supporting student loans to ensure that you wouldn’t have to clip your wings to soar high in your academic career.
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Let’s face it; essay writing doesn’t have many fans among students, more so if it’s a death penalty essay. They are long, they are boring, and demand too much research, time and effort. Hardly any awe-inspiring features right? To write death penalty essays is like being sentenced to one. They would much prefer to be facing the firing squad than having to write the essay.
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Delivering a speech and convincing the audience about your point of view at the same time need a lot of included tricks. Your listeners accept your expressed view once they find it logical and argumentative. Often people say, framing an appealing persuasive speech outline is the key to successful speech delivery.
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Literature review is a critical analysis of an authorized and accredited body of knowledge through classifications, summary, evidences, justifications and comparison of prior research studies. While writing, you must not confuse your review paper with an annotated bibliography.
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In order to present yourself as a convincing presenter of a subject and also to produce a debatable speech on it, you must understand the debate topic inside out. You should be able to do an analysis on the subject and critically think about it at any point in time.
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Narrative essay is one of the genres that allow students to come up with their creative side and imaginations that can make the composition an interesting read. One must take note of the fact that the task of writing narrative essays is to be done with care, dedication and creativity. In case you get stuck on your essay paper and lack idea on how to proceed with the task, then remember there are custom essay writing experts available these days. They can help you come up with fruitful ideas to give your intellect a creative boost. There are certain tips and helpful suggestions that can assist an individual in composing narrative essays par excellence. Take a close look at the following suggestions and expect to come up with flawless essays offering brilliant paper quality.
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Oration, that conveys some specific data, comes under the category of informative speech. While delivering the lecture, your primary target must be assisting the listeners to comprehend an intricate subject through your verbal communication. Never let your topic to render disappointment amongst your addressees. Otherwise, chances prevail that they may not pay attention to your speech.
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An abstract is important because it shows the readers if this piece of writing will serve to fulfill their purpose of studying the subject. The quality of the abstract decides whether any reader will go further with the paper or not. Thus, the main goal of any writer would be to make the abstract useful for the readers
An abstract is important because it shows the readers if this piece of writing will serve to fulfill their purpose of studying the subject. The quality of the abstract decides whether any reader will go further with the paper or not. Thus, the main goal of any writer would be to make the abstract useful for the readers
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167 attention grabbing persuasive speech topics to ensure academic successlevibaxter
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Your speech seems flawless only if you are orating on a topic that never makes you anxious. Once you become apprehensive, your logic and arguments go baseless. You can never again stream your thoughts in the direction you want it to be.
The entire sermon sounds enthusiastic and enjoyable if you speak like a spontaneously without any incoherent discourse. You can also research on the topic thoroughly if it is from your zone of interests or it is something that you like learning about.
Quick tips for choosing a topic for argumentative essayslevibaxter
An argumentative essay requires you to put forth an argument on a topic, collect and present evidence to support the argument. Considered as an analytical piece of writing, an argumentative essay can be a tough nut to crack. However, once you finalize on a topic and conduct the necessary background research, putting words to paper following a rational stream of arguments becomes much easier. These kinds of essays require a lot of in-depth research and a careful perusal of all the perspectives or arguments for and against a particular stance. Empirical evidence, such as statistics and results of previous research, is indispensable when composing a winning argumentative essay. Since choosing a topic for your argumentative essay is the first hurdle that you have to overcome, we have collected some ideas and topics that you might be able to put to use for your assignment to impress your professors and secure superb grades!
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Most of the academic institutes ask their students to choose the essay topic on their own. While it may sound liberating, it does put some additional pressure on the students.
To pick the suitable topic for your essay, you need to understand the nature of the essay you are asked to draft. Whether it’s a narrative essay, an analysis essay, a compare and contrast essay, or an expository essay – your essay topic needs to be picked according to the essay type.
Since you are given a free hand in picking the essay topic, you can pick a topic from the area where you are excel. Also, keep in mind that the topic should be relevant to your curriculum and gives you enough opportunity to explore.
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A college essay can be anything starting from a narrative essay to a persuasive one. Let us have an idea about the distinct genres of essay writing to learn writing college essays more properly, using this useful and complete Essay Writing Guide. Essays can be grouped into four major categories such as- Expository essays, descriptive essays, narrative essays and persuasive essays. These categories can be further divided into some sub-categories like informative essays, personal essays, and argumentative essays and so on.
The following factors should be considered while drafting a college essay.
• Adequate discussion on the topic
• Reference to the larger context of the issue
• Reference to more than one perspectives
• Incorporation of one’s own point of view with valid justification
Ethnobotany and Ethnopharmacology:
Ethnobotany in herbal drug evaluation,
Impact of Ethnobotany in traditional medicine,
New development in herbals,
Bio-prospecting tools for drug discovery,
Role of Ethnopharmacology in drug evaluation,
Reverse Pharmacology.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
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Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
Students, digital devices and success - Andreas Schleicher - 27 May 2024..pptxEduSkills OECD
Andreas Schleicher presents at the OECD webinar ‘Digital devices in schools: detrimental distraction or secret to success?’ on 27 May 2024. The presentation was based on findings from PISA 2022 results and the webinar helped launch the PISA in Focus ‘Managing screen time: How to protect and equip students against distraction’ https://www.oecd-ilibrary.org/education/managing-screen-time_7c225af4-en and the OECD Education Policy Perspective ‘Students, digital devices and success’ can be found here - https://oe.cd/il/5yV
TESDA TM1 REVIEWER FOR NATIONAL ASSESSMENT WRITTEN AND ORAL QUESTIONS WITH A...
Solving cubic equations with the help of factor theorem
1. Introduction
Learn to Solve Cubic Equations
In mathematical terms, all cubic equations have
either one root or three real roots. The general
cubic equation is,
ax3+ bx2+ cx+d= 0
The coefficients of a, b, c and d are real or
complex numbers with a not equals to zero
(a ≠ 0). It must have the term x3 in it, or else it
will not be a cubic equation.
2. Introduction
The coefficients of a, b, c and d are real or
complex numbers with a not equals to zero
(a ≠ 0). It must have the term x3 in it, or else it
will not be a cubic equation. But any or all of b, c
and d can be zero.
The examples of cubic equations are,
No 1, x3+ 3a3+ 3 a2 + a3– b=0
No 2, 4x3+ 57=0
No 3, x3+ 9x=0
3.
4. Strategy to Solve Cubic
Equation
Unlike quadratic equation which may have no
real solution; a cubic equation always has at
least one real root. The prior strategy of solving
a cubic equation is to reduce it to a quadratic
equation, and then solve the quadratic by usual
means, either by factorizing or using a formula.
Always try to find the solution of cubic
equations with the help of the general equation,
ax3+ bx2+ cx+d= 0
5. Strategy to Solve Cubic
Equation
A cubic equation should, therefore, must be re-
arranged into its standard form,
For example,
x2+ 4x-1 = 6/x
6. Strategy to Solve Cubic
Equation
Step 1
You can see the equation is not written in
standard form, you need to multiply the ‘x’ to
eliminate the fraction and get cubic equation,
after doing so, you will end up with
x3+ 4x2– x = 6
7. Strategy to Solve Cubic
Equation
Step 2
Then you subtract 6 from both sides in order to
get ‘0’ on the right side, so you will come up,
x3+ 4x2– x- 6 = 0
8. Solving Cubic Equations with
the help of Factor Theorem
What is factor theorem? If you divide a
polynomial p(x) by a factor x – a of that
polynomial, then, you will end up with zero as
the remainder,
p(x) = (x – a)q(x) + r(x)
9. Solving Cubic Equations with
the help of Factor Theorem
If x – a is indeed a factor of p(x), then the
remainder after division by x – a will be zero.
p(x) = (x – a)q (x)
Here is a problem,
x3– 5x2– 2x+24 = 0
With x= – 2 a solution.
10. Solving Cubic Equations with
the help of Factor Theorem
If x – a is indeed a factor of p(x), then the
remainder after division by x – a will be zero.
p(x) = (x – a)q (x)
Here is a problem,
x3– 5x2– 2x+24 = 0
With x= – 2 a solution.
11. Solving Cubic Equations with
the help of Factor Theorem
Factor theorem says that if x = – 2 is a solution
of this equation, then x+2 is a factor of this
whole expression.
12. Solving Cubic Equations with
the help of Factor Theorem
Step 1
First, you need to look at the coefficients of the
original cubic equation, which are 1, -5, -
2 and 24.
13. Solving Cubic Equations with
the help of Factor Theorem
Step 2
Now multiply number (1) that just brought
down, by the known root -2, as a result is -2, you
mention the result in the other line, like
14. Solving Cubic Equations with
the help of Factor Theorem
Step 3
The numbers in the second column are added,
so giving us,
15. Solving Cubic Equations with
the help of Factor Theorem
Step 4
Then recently written number 7 is multiplied by
the known root, – 2,
As 14 comes as a result, you need to write it
down on the second row over the line,
16. Solving Cubic Equations with
the help of Factor Theorem
Step 5
Like previously the numbers in this column
added, (14 – 2 = 12)
17. Solving Cubic Equations with
the help of Factor Theorem
Step 6
And you need to go on with the process,
18. Solving Cubic Equations with
the help of Factor Theorem
Step 7
When you have zero at the bottom row, it gives
the confirmation that x = – 2 is a root of the
original cubic. At this stage, you got the first
three numbers in the bottom row as the
coefficients in the quadratic,
x2– 7x+12
Hence, you reduced your cubic to,
19. Solving Cubic Equations with
the help of Factor Theorem
(x+2)(x2 – 7x + 12) =0
Step 8
After applying the quadratic term, the equation
comes like this,
(x +2) (x – 3) (x – 4) = 0
Resulting, you get the solution as x = -2 or 3 or 4.
20. Solving Cubic Equations with
the help of Factor Theorem
Another Example:
The equation is,
x3– 7x-6=0
Step 1 You can simply try x = – 1, after putting
the value of x, you will get,
(-1)3 – 7(-1) -6
21. Solving Cubic Equations with
the help of Factor Theorem
Step 2
After applying the synthetic division, like above
example, you will take the coefficients of the
original cubic equation, which are 1, 0, -7 and -6,
you need to write down the know root x = -1 to
the right of the vertical line, giving us,
22. Solving Cubic Equations with
the help of Factor Theorem
Step 3
Multiply the brought down number 1 by the
known root x = -1, and put down the result (-
1) at the second row, like this,
23. Solving Cubic Equations with
the help of Factor Theorem
Step 4
The numbers of the second column are added to
the first column, giving us,
24. Solving Cubic Equations with
the help of Factor Theorem
Step 5
As you add more numbers to the second column
by following the synthetic division process, you
will come with,
25. Solving Cubic Equations with
the help of Factor Theorem
Step 6
Hence, the cubic reduced to quadratic,
(x+1)(x2-x- 6) =0
The factorized result is,
(x +1)(x – 3)(x + 2) = 0
You can get three solutions to the cubic
equation are x = -2, -1 or 3
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