This document provides examples and steps for solving various types of equations beyond linear equations, including:
1) Polynomial equations solved by factoring
2) Equations with radicals where radicals are eliminated by raising both sides to a power
3) Equations with rational exponents where both sides are raised to the reciprocal power
4) Equations quadratic in form where an algebraic substitution is made to transform into a quadratic equation
5) Absolute value equations where both positive and negative solutions must be considered.
* Solve equations involving rational exponents
* Solve equations using factoring
* Solve equations with radicals and check the solutions
* Solve absolute value equations
* Solve other types of equations
* Solve equations in one variable algebraically.
* Solve a rational equation.
* Find a linear equation.
* Given the equations of two lines, determine whether their graphs are parallel or perpendicular.
* Write the equation of a line parallel or perpendicular to a given line.
* Solve quadratic equations by factoring.
* Solve quadratic equations by the square root property.
* Solve quadratic equations by completing the square.
* Solve quadratic equations by using the quadratic formula.
* Solve equations involving rational exponents
* Solve equations using factoring
* Solve equations with radicals and check the solutions
* Solve absolute value equations
* Solve other types of equations
* Solve equations in one variable algebraically.
* Solve a rational equation.
* Find a linear equation.
* Given the equations of two lines, determine whether their graphs are parallel or perpendicular.
* Write the equation of a line parallel or perpendicular to a given line.
* Solve quadratic equations by factoring.
* Solve quadratic equations by the square root property.
* Solve quadratic equations by completing the square.
* Solve quadratic equations by using the quadratic formula.
* Factor the greatest common factor of a polynomial.
* Factor a trinomial.
* Factor by grouping.
* Factor a perfect square trinomial.
* Factor a difference of squares.
* Factor the sum and difference of cubes.
* Factor expressions using fractional or negative exponents.
Expresiones algebraicas, adición y sustracción de expresiones algebraicas, multiplicación y división de expresiones algebraicas, productos notables, fraccionario de productos notables
* Evaluate square roots.
* Use the product rule to simplify square roots.
* Use the quotient rule to simplify square roots.
* Add and subtract square roots.
* Rationalize denominators.
* Use rational roots.
* Model exponential growth and decay
* Use Newton's Law of Cooling
* Use logistic-growth models
* Choose an appropriate model for data
* Express an exponential model in base e
* Construct perpendicular and angle bisectors
* Use bisectors to solve problems
* Identify the circumcenter and incenter of a triangle
* Use triangle segments to solve problems
* Factor the greatest common factor of a polynomial.
* Factor a trinomial.
* Factor by grouping.
* Factor a perfect square trinomial.
* Factor a difference of squares.
* Factor the sum and difference of cubes.
* Factor expressions using fractional or negative exponents.
Expresiones algebraicas, adición y sustracción de expresiones algebraicas, multiplicación y división de expresiones algebraicas, productos notables, fraccionario de productos notables
* Evaluate square roots.
* Use the product rule to simplify square roots.
* Use the quotient rule to simplify square roots.
* Add and subtract square roots.
* Rationalize denominators.
* Use rational roots.
* Model exponential growth and decay
* Use Newton's Law of Cooling
* Use logistic-growth models
* Choose an appropriate model for data
* Express an exponential model in base e
* Construct perpendicular and angle bisectors
* Use bisectors to solve problems
* Identify the circumcenter and incenter of a triangle
* Use triangle segments to solve problems
* Identify, write, and analyze conditional statements
* Write the inverse, converse, and contrapositive of a conditional statement
* Write a counterexample to a fake conjecture
* Find the distance between two points
* Find the midpoint of two given points
* Find the coordinates of an endpoint given one endpoint and a midpoint
* Find the coordinates of a point a fractional distance from one end of a segment
* Connect functions to their graphs
* Graph piecewise-defined functions
* Graph absolute value functions
* Graph greatest-integer functions
* Interpret graphs
* Use the vertical line test to determine a function
* Connect functions to their graphs
* Graph piecewise-defined functions
* Graph absolute value functions
* Graph greatest-integer functions
* Interpret graphs
* Use the vertical line test to determine a function
* Introduce functions and function notation
* Develop skills in constructing and interpreting the graphs of functions
* Learn to apply this knowledge in a variety of situations
* Recognize graphs of common functions.
* Graph functions using vertical and horizontal shifts.
* Graph functions using reflections about the x-axis and the y-axis.
* Graph functions using compressions and stretches.
* Combine transformations.
* Identify intervals on which a function increases, decreases, or is constant
* Use graphs to locate relative maxima or minima
* Test for symmetry
* Identify even or odd functions and recognize their symmetries
* Understand and use piecewise functions
* Determine whether a relation or an equation represents a function.
* Evaluate a function.
* Use the vertical line test to identify functions.
* Identify the domain and range of a function from its graph
* Identify intercepts from a function’s graph
* Solve counting problems using the Addition Principle.
* Solve counting problems using the Multiplication Principle.
* Solve counting problems using permutations involving n distinct objects.
* Solve counting problems using combinations.
* Find the number of subsets of a given set.
* Solve counting problems using permutations involving n non-distinct objects.
* Use summation notation.
* Use the formula for the sum of the first n terms of an arithmetic series.
* Use the formula for the sum of the first n terms of a geometric series.
* Use the formula for the sum of an infinite geometric series.
* Solve annuity problems.
* Find the common ratio for a geometric sequence.
* List the terms of a geometric sequence.
* Use a recursive formula for a geometric sequence.
* Use an explicit formula for a geometric sequence.
* Find the common difference for an arithmetic sequence.
* Write terms of an arithmetic sequence.
* Use a recursive formula for an arithmetic sequence.
* Use an explicit formula for an arithmetic sequence.
* Write the terms of a sequence defined by an explicit formula.
* Write the terms of a sequence defined by a recursive formula.
* Use factorial notation.
* Identify characteristics of each type of conic section
* Identify a conic section from its equation in general form
* Identifying the eccentricities of each type of conic section
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.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
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 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
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.
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
CLASS 11 CBSE B.St Project AIDS TO TRADE - INSURANCE
1.6 Other Types of Equations
1. 1.6 Other Types of Equations
Chapter 1 Equations and Inequalities
2. Concepts and Objectives
⚫ Objectives for this section are:
⚫ Solve polynomial equations by factoring
⚫ Solve equations with radicals and check the solutions
⚫ Solve equations with rational exponents
⚫ Solve equations that are quadratic in form
⚫ Solve absolute value equations
3. Solving by Factoring
⚫ Some polynomial equations of degree 3 or higher can be
solved by moving all terms to one side (thus setting the
equation equal to 0), factoring the result, and setting
each factor to 0.
⚫ Usually, we will be looking for the GCF of each term.
⚫ Example: Solve by factoring: =
4 2
3 27
x x
4. Solving by Factoring (cont.)
⚫ Example: Solve by factoring:
Since 3 is a factor on both sides, we can divide both
sides by 3 in order to simplify each side. Even though x2
is also a factor, we generally do not divide by variables.
=
4 2
3 27
x x
( )
4 2
4 2
4 2
2 2
3 27
9
9 0
9 0
x x
x x
x x
x x
=
=
− =
− =
( )( )
2
3 3 0
0, 3,3
x x x
x
+ − =
= −
5. Solving by Grouping
⚫ If a polynomial consists of 4 terms, we can sometimes
solve it by grouping.
⚫ Grouping procedures require factoring the first two
terms, and then factoring the last two terms. If the
factors in the parentheses are identical, then the
expression can be factored by grouping.
6. Solving by Grouping
⚫ Example: Solve
Solutions are ‒1, 3, ‒3
3 2
9 9 0
x x x
+ − − =
( ) ( )
( )( )
( )( )( )
3 2
2
2
9 9 0
1 9 1 0
1 9 0
1 3 3 0
x x x
x x x
x x
x x x
+ − − =
+ − + =
+ − =
+ − + =
difference of squares!
7. Rational Exponent Equations
⚫ Recall that a rational exponent indicates a power in the
numerator and a root in the denominator.
⚫ There are multiple ways of writing an expression with a
rational exponent:
⚫ If we are given an equation in which a variable is raised
to a rational exponent, the simplest way to remove the
exponent on x is by raising both sides of the equation to
a power that is the reciprocal of the exponent (flip the
fraction).
( ) ( )
/ 1/
m
m
m n n m
n n
a a a a
= = =
8. Rational Exponent Equations
⚫ Example: Solve
The reciprocal of is , so
⚫ To enter this on your calculator (or Desmos), use the ›
key (called a “caret”).
5/4
32
x =
5
4
4
5
( ) ( )
4/5 4/5
5/4
32
16
x
x
=
=
9. Rational Exponent Equations
⚫ Example: Solve
We can now use the zero-factor property:
3/4 1/2
3x x
=
( )
3/4 1/2
3/4 2/4
2/4 1/4
3 0
3 0
3 1 0
x x
x x
x x
− =
− =
− =
10. Rational Exponent Equations
⚫ Example: Solve
Solutions are 0 and
3/4 1/2
3x x
=
2/4
0
0
x
x
=
=
1/4
1/4
1/4
4
3 1 0
3 1
1
3
1 1
3 81
x
x
x
x
− =
=
=
= =
1
81
11. Power Property
⚫ Note: This does not mean that every solution of Pn = Qn
is a solution of P = Q.
⚫ We use the power property to transform an equation
that is difficult to solve into one that can be solved more
easily. Whenever we change an equation, however, it is
essential to check all possible solutions in the original
equation.
If P and Q are algebraic expressions, then every
solution of the equation P = Q is also a solution of
the equation Pn = Qn, for any positive integer n.
12. Solving Radical Equations
⚫ Step 1 Isolate a radical on one side of the equation.
⚫ Step 2 Raise each side of the equation to a power that is
the same as the index of the radical to eliminate the
radical.
⚫ If the equation still contains a radical, repeat steps 1
and 2 until no radicals remain.
⚫ Step 3 Solve the resulting equation.
⚫ Step 4 Check each proposed solution in the original
equation.
14. Solving Radical Equations (cont.)
⚫ Example: Solve − + =
4 12 0
x x
= +
4 12
x x
= +
2
4 12
x x
− − =
2
4 12 0
x x
( )( )
− + =
6 2 0
x x
= −
6, 2
x
15. Solving Radical Equations (cont.)
⚫ Example: Solve
Check:
Solution: {6}
− + =
4 12 0
x x
4 12
x x
= +
2
4 12
x x
= +
2
4 12 0
x x
− − =
( )( )
6 2 0
x x
− + =
6, 2
x = −
( )
− + =
6 4 6 12 0
− =
6 36 0
− =
6 6 0
=
0 0
( )
− − − + =
2 4 2 12 0
− − =
2 4 0
− − =
2 2 0
−
4 0
17. Solving Radical Equations (cont.)
⚫ Example: Solve + − + =
3 1 4 1
x x
( )
2
4 1
x + +
( )
2
2
3 1 4 1
3 1 4 2 4 1
2 4 2 4
2 4
4 4 4
5 0
5 0
0, 5
x x
x x x
x x
x x
x x x
x x
x x
x
+ = + +
+ = + + + +
− = +
− = +
− + = +
− =
− =
=
18. Solving Radical Equations (cont.)
⚫ Example: Solve
Check:
Solution: {5}
+ − + =
3 1 4 1
x x
( )
2
2
3 1 4 1
3 1 4 2 4 1
2 4 2 4
2 4
4 4 4
5 0
5 0
0, 5
x x
x x x
x x
x x
x x x
x x
x x
x
+ = + +
+ = + + + +
− = +
− = +
− + = +
− =
− =
=
( )+ − + =
3 0 1 0 4 1
− =
1 4 1
− =
1 2 1
−
1 1
( )+ − + =
3 5 1 5 4 1
− =
16 9 1
− =
4 3 1
=
1 1
19. Solving Radical Equations (cont.)
You can also use the table on your graphing calculator to
check your answers.
⚫ Enter the left side of original equation into Y1 and the
right side of the equation into Y2 (use y¡ to create a
radical)
⚫ Go to the table (ys) and go to your first x-value. If
it is a solution, Y1 should equal Y2. If it doesn’t, it’s not a
solution.
20. Quadratic in Form
⚫ An equation is said to be quadratic in form if it can be
written as
where a 0 and u is some algebraic expression.
⚫ To solve this type of equation, substitute u for the
algebraic expression, solve the quadratic expression for
u, and then set it equal to the algebraic expression and
solve for x. Because we are transforming the equation,
you will still need to check any proposed solutions against
the original equation.
+ + =
2
0
au bu c
21. Quadratic in Form (cont.)
⚫ Example: Solve ( ) ( )
− + − − =
2 3 1 3
1 1 12 0
x x
⅔ is two times ⅓
This is what makes it
quadratic in form.
22. Quadratic in Form (cont.)
⚫ Example: Solve
Let . This makes our equation:
( ) ( )
− + − − =
2 3 1 3
1 1 12 0
x x
( )
= −
1 3
1
u x
+ − =
2
12 0
u u
( )( )
+ − =
4 3 0
u u
= −4, 3
u
23. Quadratic in Form (cont.)
⚫ Example: Solve
Let . This makes our equation:
So, and
( ) ( )
− + − − =
2 3 1 3
1 1 12 0
x x
( )
= −
1 3
1
u x
+ − =
2
12 0
u u
( )( )
+ − =
4 3 0
u u
= −4, 3
u
( )
( ) ( )
1 3
3
1/3 3
1 4
1 4
1 64
63
x
x
x
x
− = −
− = −
− = −
= −
( )
( ) ( )
1 3
3
1/3 3
1 3
1 3
1 27
28
x
x
x
x
− =
− =
− =
=
25. Quadratic in Form (cont.)
⚫ Example: Solve (cont.)
Solution: {–63, 28}
(as before, you can also use your calculator to check)
( ) ( )
− + − − =
2 3 1 3
1 1 12 0
x x
( ) ( )
− + − − =
2 3 1 3
28 1 28 1 12 0
( ) ( )
+ − =
2 3 1 3
27 27 12 0
( ) ( )
+ − =
2 1
3 3 12 0
+ − =
9 3 12 0
=
0 0
26. Absolute Value Equations
⚫ You should recall that the absolute value of a number a,
written |a|, gives the distance from a to 0 on a number
line.
⚫ By this definition, the equation |x| = 2 can be solved by
finding all real numbers at a distance of 2 units from 0.
Both of the numbers 2 and ‒2 satisfy this equation, so
the solution set is {‒2, 2}.
27. Absolute Value Equations (cont.)
⚫ The solution set for the equation must include
both a and –a.
⚫ Example: Solve
=
x a
− =
9 4 7
x
28. Absolute Value Equations (cont.)
⚫ The solution set for the equation must include
both a and –a.
⚫ Example: Solve
The solution set is
=
x a
− =
9 4 7
x
− =
9 4 7
x − = −
9 4 7
x
− = −
4 2
x − = −
4 16
x
=
1
2
x = 4
x
or
1
,4
2
30. Absolute Value Equations (cont.)
⚫ Example: Solve
Before we do anything else, we have to isolate the
absolute value expression:
51 4 15 0
x
− − =
51 4 15
1 4 3
x
x
− =
− =
1 4 3
4 2
1
2
x
x
x
− =
− =
= −
or
1 4 3
4 4
1
x
x
x
− = −
− = −
=
1
, 1
2
−
31. For Next Class
⚫ Section 1.6 in MyMathLab
⚫ Quiz 1.6 in Canvas
⚫ Optional: Read section 2.1 in your textbook
Remember that the deadline for the homework and quiz is
Sunday at 11:59 pm!