1) The document discusses factoring quadratic equations of the form ax^2 + bx + c = 0.
2) It provides three examples of factoring and solving quadratic equations: 2x^2 - x - 6 = 0, 10x^2 + 3x - 1 = 0, and 3x^2 - x - 2 = 0.
3) The method involves writing the equation in standard form, factoring the left side into two binomials, setting each binomial equal to zero using the zero product property, and solving for the roots.
anyone who need my power point presentation, manual and detailed lesson plan for Algebra for Grade 7. you can purchase it. just text me at 09192920646 or email me at alexmrrn@yahoo.com
anyone who need my power point presentation, manual and detailed lesson plan for Algebra for Grade 7. you can purchase it. just text me at 09192920646 or email me at alexmrrn@yahoo.com
This is a presentation by Dada Robert in a Your Skill Boost masterclass organised by the Excellence Foundation for South Sudan (EFSS) on Saturday, the 25th and Sunday, the 26th of May 2024.
He discussed the concept of quality improvement, emphasizing its applicability to various aspects of life, including personal, project, and program improvements. He defined quality as doing the right thing at the right time in the right way to achieve the best possible results and discussed the concept of the "gap" between what we know and what we do, and how this gap represents the areas we need to improve. He explained the scientific approach to quality improvement, which involves systematic performance analysis, testing and learning, and implementing change ideas. He also highlighted the importance of client focus and a team approach to quality improvement.
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.
The Indian economy is classified into different sectors to simplify the analysis and understanding of economic activities. For Class 10, it's essential to grasp the sectors of the Indian economy, understand their characteristics, and recognize their importance. This guide will provide detailed notes on the Sectors of the Indian Economy Class 10, using specific long-tail keywords to enhance comprehension.
For more information, visit-www.vavaclasses.com
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 Art Pastor's Guide to Sabbath | Steve ThomasonSteve Thomason
What is the purpose of the Sabbath Law in the Torah. It is interesting to compare how the context of the law shifts from Exodus to Deuteronomy. Who gets to rest, and why?
How to Create Map Views in the Odoo 17 ERPCeline George
The map views are useful for providing a geographical representation of data. They allow users to visualize and analyze the data in a more intuitive manner.
3. Example 1. Solve 2x2 – x – 6 = 0
Steps Solution
1. Write in standard form. 2x2 – x – 6 = 0
2. Factor the quadratic expression on the
left side of the equation using the
following process.
a. Multiply (a) (c)
b. Find the factors of -12 that gives you
the sum of the value of b which is -1.
c. Copy the expression and replace the
middle term as the value obtain in step b.
d. Factor using the groupings
(2).(-6) = -12
-12 = (-4).(3)
2x2 – x – 6 = 0
2x2 – 4x + 3x - 6 = 0
(2x2 – 4x) + (3x – 6) = 0
2x (x – 2) + 3 (x – 2) = 0
(2x + 3) (x – 2) = 0
4. 3. Use the Zero Product Property. (2x + 3) (x – 2) = 0
So, (2x + 3 ) = 0 and (x – 2) = 0
4. Solve for the roots 2x + 3 = 0 and x – 2 = 0
2x = -3 and x = 2
𝟐𝒙
𝟐
=
−𝟑
𝟐
x = -
𝟑
𝟐
Therefore, the roots of x2 + 3x + 2 = 0 are x = -1 and x = -2.
5. Example 2. Solve for the roots of 10x2 + 3x - 1 = 0
Steps Solution
1.Write in standard form 10x2 + 3x – 1 = 0
2. Factor the expression on the left side.
a. Multiply (a). (c)
b. Factors of -10 that gives the sum of b
which is 3.
c. Copy the expression and replace the
middle term.
d. Factor using groupings.
(10) . (-1) = -10
(3)= (5) . (-2)
10x2 + 3x – 1 = 0
10x2 + 5x – 2x – 1 = 0
(10x2 + 5x) – (2x – 1) = 0
5x(2x + 1) – 1(2x + 1) = 0
(5x – 1) (2x + 1) = 0
6. 3. Use the Zero Product Property
.
(5x - 1) (2x + 1) = 0
So, 5x – 1 = 0 and 2x + 1 = 0
4. Solve for the roots. 5x – 1 = 0 and 2x + 1 = 0
5x = 1 and 2x = -1
x =
𝟏
𝟓
and x = -
𝟏
𝟐
So, the roots of 10x2 + 3x – 1 = 0 are x =
𝟏
𝟓
and x = -
𝟏
𝟐
7. Example 3. Find the roots of 3x2 = x + 2
Steps Solution
1.Write in standard form 3x2 – x – 2 = 0
2. Factor the left side of the expression.
a. Multiply (a) . (c) (3). (-2) = -6
b. Find the factors of 6 that gives you
the sum of the value of b which is -1.
c. Copy the expression and replace the
middle term by the value obtain in step
b.
d. Factor using the method of
groupings.
(-1) = (-3) . (2)
3x2 – x – 2 = 0
3x2 – 3x + 2x – 2 = 0
(3x2 – 3x) + (2x – 2) = 0
3x(x – 1) + 2(x – 1) = 0
(3x + 2) (x – 1) = 0
8. 3. Use the Zero Product Property (3x + 2) (x – 1) = 0
(3x + 2) = 0 and (x – 1) = 0
4. Solve for the roots. 3x + 2 = 0 and (x – 1) = 0
3x = -2 and x = 1
3𝑥
3
=
−2
3
x = -
𝟐
𝟑
Therefore, the roots of 3x2 = x + 2 are -
𝟐
𝟑
and 1.
