1. The document discusses various methods for solving different types of equations including: polynomial equations using factoring, quadratic equations using the quadratic formula, rational equations by clearing denominators, radical equations by squaring both sides, and absolute value equations geometrically.
2. Specific examples are provided to demonstrate how to solve equations involving polynomials, rationals, radicals, powers, and absolute values. Steps include factoring, using the quadratic formula, clearing denominators, isolating and squaring radicals, and raising both sides to appropriate powers.
3. The document also discusses finding zeros of rational functions and domains by determining where the numerator or denominator is equal to zero.
College algebra real mathematics real people 7th edition larson solutions manualJohnstonTBL
College Algebra Real Mathematics Real People 7th Edition Larson Solutions Manual
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Quadratic Equations
In One Variable
1. Quadratic Equation
an equation of the form
ax2 + bx + c = 0
where a, b, and c are real numbers
2.Types of Quadratic Equations
Complete Quadratic
3x2 + 5x + 6 = 0
Incomplete/Pure Quadratic Equation
3x2 - 6 = 0
3.Solving an Incomplete Quadratic
4.Example 1. Solve: x2 – 4 = 0
Solution:
x2 – 4 = 0
x2 = 4
√x² = √4
x = ± 2
5.Example 2. Solve: 5x² - 11 = 49
Solution:
5x² - 11 = 49
5x² = 49 + 11
5x² = 60
x² = 12
x = ±√12
x = ±2√3
6.Solving Quadratic Equation
7.By Factoring
Place all terms in the left member of the equation, so that the right member is zero.
Factor the left member.
Set each factor that contains the unknown equal to zero.
Solve each of the simple equations thus formed.
Check the answers by substituting them in the original equation.
8.Example: x² = 6x - 8
Solution:
x² = 6x – 8
x² - 6x + 8 = 0
(x – 4)(x – 2) = 0
x – 4 = 0 | x – 2 = 0
x = 4 x = 2
9.By Completing the Square
Write the equation with the variable terms in the left member and the constant term in the right member.
If the coefficient of x² is not 1, divide every term by this coefficient so as to make the coefficient of x² equal to 1.
Take one-half the coefficient of x, square this quantity, and add the result to both members.
Find the square root of both members, placing a ± sign before the square root of the right member.
Solve the resulting equation for x.
10.Example: x² - 8x + 7 = 0
11.By Quadratic Formula
Example: 3x² - 2x - 7 = 0
12.Solve the following:
1. x² - 15x – 56 = 0
2. 7x² = 2x + 6
3. 9x² - 3x + 8 = 0
4. 8x² + 9x -144 = 0
5. 2x² - 3 + 12x
13.Activity:
Solve the following quadratic formula.
By Factoring By Quadratic Formula
1. x² - 5x + 6 = 0 1. x² - 7x + 6 = 0
2. 3 x² = x + 2 2. 10 x² - 13x – 3 = 0
3. 2 x² - 11x + 12 = 0 3. x (5x – 4) = 2
By Completing the Square
1. x² + 6x + 5 = 0
2. x² - 8x + 3 = 0
3. 2 x² + 3x – 5 = 0
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!
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
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.
Instructions for Submissions thorugh G- Classroom.pptx
1.3 solving equations t
1. Solving Equations
Example A. 3x2 – 2x = 8
We solve polynomial equations by factoring.
Set one side to 0, 3x2 – 2x – 8 = 0 factor this
(3x + 4)(x – 2) = 0 extract answers
so x = –4/3, 2
the roots for the equation ax2 + bx + c = 0 are
x =
–b ± b2 – 4ac
2a
A “root” is a solution for
the equation “# = 0”.
For 2nd degree or quadratic equations, we may also
use the Quadratic Formula (QF):
For 3x2 – 2x – 8 = 0, a = 3, b = –2, c = –8,
so b2 – 4ac = 100, or b2 – 4ac = 10
Hence x = (2 – 10)/6 or (2 + 10)/6
or x = –4/3, 2
2. Rational Equations
Solve rational equations by clearing all denominators
using the LCD.
Example B. Solve
LCD = (x – 2)(x + 1), multiply the LCD to both sides
of the equation:
]
2(x + 1) = 4(x – 2) + 1*(x + 1)(x – 2)
2x + 2 = 4x – 8 + x2 – x – 2
2x + 2 = x2 + 3x – 10
0 = x2 + x – 12
0 = (x + 4)(x – 3) x = -4, 3
Both are good.
x – 2
2 =
x + 1
4
+ 1
(x + 1) (x – 2) (x + 1)(x – 2)
x – 2
2 = x + 1
4 + 1
(x – 2)(x + 1) * [
3. Power Equations
To solve a power equation, take the reciprocal power,
so if xR = c, xp/q = c
x = (±)cq/por
Equations of the Form xp/q = c
then x = (±)c1/R
Example F. Solve (2x – 3)3/2 = -8
Raise both sides to 2/3 power.
(2x – 3)3/2 = -8 [(2x – 3)3/2]2/3 = (-8)2/3
(2x – 3) = 4
2x = 7 x = 7/2
Since x = 7/2 doesn't work because 43/2 = -8,
there is no solution.
4. Radical Equations
Solve radical equations by squaring both sides to
remove the square root. Do it again if necessary.
Reminder: (A ± B)2 = A2 ± 2AB + B2
Example G. Solve
x + 4 = 5x + 4 square both sides;
(x + 4)2 = (5x + 4 )2
x + 2 * 4 x + 16 = 5x + 4 isolate the radical;
8x = 4x – 12 divide by 4;
2x = x – 3 square again;
( 2x)2 = (x – 3)2
4x = x2 – 6x + 9
0 = x2 – 10x + 9
0 = (x – 9)(x – 1)
x = 9, x = 1 Only 9 is good.
5. The geometric meaning of “|x – y|” or “|y – x|” is
“the distance between x and y”.
So the absolute value equation “| x – c | = r” is claiming
that “the distance from x to c is r”. Hence x = c ± r.
Absolute Value Equations
Example G. Draw and solve for x geometrically
if |x – 7| = 12.
The equation asks for the locations of x's that are
12 units away from 7.
c
r
x = c + r
So to the right x = 7 + 12 = 19.
and to the left x = 7 – 12 = – 5.
7
1212
x = 19x = – 5
r
x = c – r
6. Zeroes and Domains
Example I.
For a reduced fractional formula f = ,
the zeroes of f are where N = 0
the domain of f are where “D = 0”
N
D
x2 – 4.Find the zeros and the domain f = x2 – 1
The zeros of f are where x2 – 1 = 0 or x = ±1.
The domain of f are
“all numbers x2 – 4 = 0, or all numbers except x = ±2”.
7. A. Solve the following equations by factoring.
5. x2 – 3x = 10
9. x3 – 2x2 = 0
6. x2 = 4
7. 2x(x – 3) + 4 = 2x – 4
10. 2x2(x – 3) = –4x
8. x(x – 3) + x + 6 = 2x2 + 3x
1. x2 – 3x – 4 = 0 2. x2 – 2x – 15 = 0 3. x2 + 7x + 12 = 0
4. –x2 – 2x + 8 = 0
11. 4x2 = x4
12. 7x2 = –4x3 – 3x 13. 5 = (x + 2)(2x + 1)
14. (x + 1)2 = x2 + (x – 1)2 15. (x + 3)2 – (x + 2)2 = (x + 1)2
B. Solve the following equations by the quadratic formula.
If the answers are not real numbers, just state so.
1. x2 – x + 1 = 0 2. x2 – x – 1 = 0
3. x2 – 3x – 2 = 0 4. x2 – 2x + 3 = 0
5. 2x2 – 3x – 1 = 0 6. 3x2 = 2x + 3
Equations
8. Rational Equations
7. 1
x
+
1
x – 1
=
5
6
8. 1
x
+
1
x + 2
=
3
4
9. 2
x
+ 1
x + 1
= 3
2
10. + 5
x + 2
= 22
x – 1
11. – 1
x + 1
= 3
2
12.6
x + 2
– 4
x + 1
= 11
x – 2
x
6 3
1 2
3
5
2
3–+ = x1. x
4 6
–3 1
8
–5
– 1– = x2.
x
4 5
3 2
10
7
4
3+– = x3. x
8 12
–5 7
16
–5
+ 1+ = x
(x – 20) = x – 3
100
30
100
205.
(x + 5) – 3 = (x – 5)
100
25
100
206.
C. Clear the denominators of the equations, then solve.
9. Radical Equations and Power Equations
D. Isolate one radical if needed, square both sides, do it again
if necessary, to solve for x. Check your answers.
1. x – 2 = x – 4 2. x + 3 = x + 1
3. 2x – 1 = x + 5 4. 4x + 1 – x + 2 = 1
5. x – 2 = x + 3 – 1 6. 3x + 4 = 3 – x – 1
7. 2x + 5 = x + 4 8. 5 – 4x – 3 – x = 1
E. Solve by raising both sides to an appropriate power.
No calculator.
1. x –2 = 1/4 2. x –1/2 = 1/4
3. x –3 = –8 4. x –1/3 = –8
5. x –2/3 = 4 6. x –3/2 = 8
7. x –2/3 = 1/4 8. x –3/2 = – 1/8
9. x 1.5 = 1/27 10. x 1.25 = 32
11. x –1.5 = 27 12. x –1. 25 = 1/32
10. F. Solve for x.
1. Is it always true that I+x| = x? Give reason for your answer.
2. Is it always true that |–x| = x? Give reason for your answer.
Absolute Value Equations
3. |4 – 5x| = 3 4. |3 + 2x| = 7 5. |–2x + 3| = 5
6. |4 – 5x| = –3 7. |2x + 1| – 1= 5 8. 3|2x + 1| – 1= 5
9. |4 – 5x| = |3 + 2x|
11. |4 – 5x| = |2x + 1| 12. |3x + 1| = |5 – x|
10. |–2x + 3|= |3 – 2x|
Solve geometrically for x. Draw the solution.
13. |x – 2| = 1 14. |3 + x| = 5 15. | –9 + x| = –7
x – 4
G. Find the zeros and the domain of the following rational
formulas. (See 2.1)
2x – 1
1. x2 – 1.
x2 – 43.
5x + 7
2.
3x + 5
x2 – x
x2 – x – 24.
x2 – 4x5.
x2 + x – 2
x2 + 2x6.
x2 + x + 2
2x2 – x – 17.
x3 + 2x
x4 – 4x8.
x3 – 8
16. |2 + x| = 1 17. |3 – x| = –5 18. | –9 – x| = 8