This document discusses solving applied problems by translating word problems into algebraic equations. It provides 5 steps for problem solving: 1) familiarize yourself with the problem, 2) translate to an equation, 3) solve the equation, 4) check the answer, and 5) state the answer clearly. It then works through examples of translating and solving various word problems into equations, such as finding lengths of pipe cut into pieces, number of photos that can be printed, apartment numbers based on a sum, angles of a triangle, and an amount needed to sell a house for after commissions.
Positional number systems represent numbers as a sequence of digits with each digit position having an associated weight. In positional systems, the value of a number is the sum of the digit values multiplied by their weights. Common positional systems include decimal, binary, octal, and hexadecimal. Converting between these systems involves grouping digits and mapping groups to new bases. Negative numbers are represented using sign-magnitude or two's complement systems.
The document discusses expressions and equations. It provides an example of calculating the total cost of ordering x pizzas from Pizza Grande using the expression "8x + 10". It then shows how to solve the equation "8x + 10 = 810" to determine that x = 100 pizzas were ordered. The document explains that equations set two expressions equal and solving an equation means finding the value of the variable that makes the equation true. It distinguishes between linear and quadratic equations.
This document contains a practice math test with 10 questions and detailed solutions. It addresses topics like finding the least common multiple, calculating percentages from a pie chart, identifying perfect squares and cubes, substitution in algebraic expressions, calculating averages, and using properties of 30-60-90 triangles to solve for lengths in a geometry problem. The practice test provides step-by-step workings to demonstrate how to arrive at the answers as well as expert shortcuts for some questions.
The document provides solutions to 10 math word problems. Question 10 asks for the length of segment AE given that segment ED is 6 units, segment AB is 10 units, and angle ECD is 60 degrees. The solution notes that triangles CDE and EAF are both 30-60-90 triangles. It is determined that EF = 4√3, BF = 2√3, and AF = 10 - 2√3. Using the properties of 30-60-90 triangles, the length of AE is calculated to be 20 - 4√3 units.
Este taller vamos a ver una rama de las matemáticas que se ocupa del estudio de las propiedades de las figuras en el plano o el espacio, incluyendo: puntos, rectas, planos etc
The document provides information about a management aptitude test and social entrepreneurship program. It discusses developing change makers and offers a free, comprehensive program in social and spiritual entrepreneurship open to all. It then provides examples of math and reasoning questions along with solutions.
This is the ultimate set of game-changer, the nuclear bomb of calculations, the Best, Just follow the rules and beat the computer
The ultimate tricks to speed up your Calculating Power
Positional number systems represent numbers as a sequence of digits with each digit position having an associated weight. In positional systems, the value of a number is the sum of the digit values multiplied by their weights. Common positional systems include decimal, binary, octal, and hexadecimal. Converting between these systems involves grouping digits and mapping groups to new bases. Negative numbers are represented using sign-magnitude or two's complement systems.
The document discusses expressions and equations. It provides an example of calculating the total cost of ordering x pizzas from Pizza Grande using the expression "8x + 10". It then shows how to solve the equation "8x + 10 = 810" to determine that x = 100 pizzas were ordered. The document explains that equations set two expressions equal and solving an equation means finding the value of the variable that makes the equation true. It distinguishes between linear and quadratic equations.
This document contains a practice math test with 10 questions and detailed solutions. It addresses topics like finding the least common multiple, calculating percentages from a pie chart, identifying perfect squares and cubes, substitution in algebraic expressions, calculating averages, and using properties of 30-60-90 triangles to solve for lengths in a geometry problem. The practice test provides step-by-step workings to demonstrate how to arrive at the answers as well as expert shortcuts for some questions.
The document provides solutions to 10 math word problems. Question 10 asks for the length of segment AE given that segment ED is 6 units, segment AB is 10 units, and angle ECD is 60 degrees. The solution notes that triangles CDE and EAF are both 30-60-90 triangles. It is determined that EF = 4√3, BF = 2√3, and AF = 10 - 2√3. Using the properties of 30-60-90 triangles, the length of AE is calculated to be 20 - 4√3 units.
Este taller vamos a ver una rama de las matemáticas que se ocupa del estudio de las propiedades de las figuras en el plano o el espacio, incluyendo: puntos, rectas, planos etc
The document provides information about a management aptitude test and social entrepreneurship program. It discusses developing change makers and offers a free, comprehensive program in social and spiritual entrepreneurship open to all. It then provides examples of math and reasoning questions along with solutions.
This is the ultimate set of game-changer, the nuclear bomb of calculations, the Best, Just follow the rules and beat the computer
The ultimate tricks to speed up your Calculating Power
The document discusses various methods for teaching calculation to primary school students. It covers teaching addition, subtraction, multiplication and division. For each operation, it provides examples of methods like using number lines, partitioning, repeated addition/subtraction, and formal written methods. The goal is to help students develop a strong foundation in understanding numbers and calculations.
This document introduces the Nikhilam Sutra, a method of Vedic mathematics for multiplication. It explains the principles and provides examples of multiplying numbers near and away from multiples of 10 using appropriate bases. The key steps are to make the numbers equal in digits, choose a base, find the differences from the base, add the numbers and differences, and multiply the differences. It also covers cases where the numbers are slightly above or below multiples and proportional methods for numbers with rational relationships. Practice problems are provided to demonstrate applying the sutra.
This document provides instructions and examples for using a math workbook on fractions. It explains how to work through the book step-by-step from less complex to more complex topics like addition, subtraction, multiplication and division of fractions. Examples are provided to demonstrate reducing fractions to a common denominator and performing operations on homogeneous and heterogeneous fractions. Time limits are given for each section to practice problems until they can be completed within the allotted time.
The document provides examples and explanations for solving problems involving patterns and functions. It presents a five-step plan for problem solving: 1) read the problem, 2) plan how to set up and solve, 3) do the work, 4) answer the question, and 5) check the answer. It then works through three examples step-by-step to find patterns in tables and write equations to describe the relationships. The homework assignment is to complete problems 1-20 on page 274.
The document provides information on numerical reasoning concepts including arithmetic progression, geometric progression, formulas, ratio and proportion problems, alligation, and mixture problems. It includes 15 multi-step word problems covering these topics and their step-by-step solutions. The problems demonstrate how to set up and solve ratios, proportions, alligation and mixture scenarios to find unknown values.
This document outlines a three day lesson plan on solving equations using properties of equality. The objectives are to solve linear equations using addition, subtraction, division, and multiplication properties. Students will learn to solve equations of the form x + b = c, ax = c, and ax + b = c. The lessons include step-by-step examples and word problems. Students will practice solving equations on their own. Additional online resources are provided for extra practice.
The document contains 31 multi-step math word problems with solutions. The problems cover a range of topics including percentages, ratios, averages, probability, geometry, and more. The level of difficulty ranges from relatively simple to more complex.
This document contains 6 problems and their solutions from a Regional Mathematical Olympiad competition in 2010. Problem 1 involves proving that the area of one triangle is the geometric mean of the areas of two other triangles in a hexagon with concurrent diagonals. Problem 2 proves that if three quadratic polynomials have a common root, then their coefficients must be equal. Problem 3 counts the number of 4-digit numbers divisible by 4 but not 8 having non-zero digits. Problem 4 finds the smallest positive integers whose reciprocals satisfy certain relationships. Problem 5 proves that the reflection of a point in a line lies on a side of a triangle. Problem 6 determines the number of values of n where an is greater than an+1 for a defined sequence
This document outlines lessons from a mathematics textbook on multiplying multi-digit numbers by two-digit numbers. It includes 7 lessons that cover multiplying by tens and hundreds, estimating products, using different problem-solving strategies like acting it out, standard algorithms for multiplying two-digit and three-digit numbers by two-digit numbers, and choosing the best strategy for a given problem. Examples and practice problems are provided for each lesson.
This document discusses different number systems such as binary, decimal, hexadecimal, and octal. It provides details on how to convert between these number systems using techniques like multiplying each bit by its place value. Examples are given for converting between the different bases to illustrate concepts like binary addition, multiplication, and representing fractions.
This document provides methods, tips, and tricks for multiplying two numbers that start with the same digit using a reference point method. It demonstrates how to solve multiplication problems by determining how much each number is from a chosen reference point, usually a multiple of 10. It shows examples of multiplying numbers using different reference points from 10 to 100 and explains how to choose an appropriate reference point based on the numbers. The key steps are to find the difference of each number from the reference point, multiply those differences and add it to the product of multiplying the numbers' sums from the reference point.
This document contains figures and images related to the digestive system. It includes over 50 labeled figures showing the organs and structures of the digestive system, including the mouth, esophagus, stomach, small intestine, large intestine, liver, pancreas and gallbladder. The figures show the locations and relationships between these organs, as well as microscopic views of tissues and diagrams of metabolic pathways.
E-learning in Medical Education and Blended Learning Approach discusses the history and applications of e-learning in medical education. It begins by explaining how education has shifted from teacher-centered to learner-centered and how e-learning can improve learning outcomes. E-learning uses technology like the internet to provide educational resources beyond the classroom. While e-learning has advantages, traditional learning still has benefits. The document then reviews the history of using computers in medical education from the 1960s onward. It describes different modes of e-learning delivery and applications in medical education today.
Brock biology-of-microorganisms-(13th-edition)Shahab Pour
This document provides an overview of microbiology. It begins with an introduction to microbiology as a science and discusses the historical discoveries that contributed to the field. It then describes techniques used to study microbes like microscopy. It explores the diversity of microbial life including bacteria, archaea, and microbial eukaryotes. The document also examines microbial cell structure, metabolism, growth, genetics, and genomics. It discusses the roles of microbes in areas like photosynthesis, biogeochemical cycles, and commercial applications. Finally, it considers microbial evolution and systematics.
Hamsters, gerbils, and ferrets are common small mammal pets. Hamsters have cheek pouches and flank glands, and require exercise wheels in their cages. Gerbils are adapted to desert environments, have a scent gland on their stomach, and are generally quiet. Ferrets are carnivores descended from European polecats, and require protein-rich commercially available diets or high-quality cat food. Both hamsters and gerbils can be handled gently by lifting behind the front or rear quarters, while ferrets are best lifted by the nape of the neck or with one hand around the body.
Edexcel as chemistry tag 2nd ed (Book answers and teachers guide)Samith Senadeera
This document is a teacher guide for the Edexcel AS Chemistry textbook. It provides answers to questions at the end of each chapter in the textbook to help students understand the concepts. For each answer, additional examiner notes are included to explain the reasoning and common mistakes to avoid. The guide also contains model answers for practice unit tests with marking points indicated. It is designed to help both teachers and students with the Edexcel AS Chemistry curriculum.
The document provides an overview of an introductory chapter on computers and C++ programming. It discusses computer systems and components, including hardware, memory, processors, and software. It then covers programming concepts like algorithms, problem solving approaches, and the C++ programming language. Specific topics include data representation in memory, compilers, object-oriented programming, and the software development lifecycle. Sample C++ code is presented and explained to demonstrate basic program structure and I/O.
This document provides an overview and instructions for an eBook format called ePUB. It discusses that support for ePUB features varies across devices. It recommends customizing settings like font, size, and layout. It notes that some eBooks include code examples that are best viewed in single column landscape mode. Images are included to mimic the print version code display. Clicking links will show the print-fidelity code image. The back button returns to the previous page.
The document is the solutions manual for a fluid mechanics textbook. It contains solutions to example problems from Chapter 1 which covers basic concepts in fluid mechanics, including definitions of different types of flows, fluid properties, forces, and units. The solutions provide concise explanations and calculations for each question with the relevant equations, assumptions, analysis, and discussions of the results.
The document describes steps for solving applied problems using linear equations:
1. Read the problem and identify what is given and what needs to be found.
2. Assign a variable to represent the unknown and write an equation relating the variable to the information given.
3. Solve the equation to find the value of the variable, which represents the answer.
It then provides examples of applying these steps to problems involving unknown numbers, sums of quantities, supplementary and complementary angles, and consecutive integers.
The document discusses various methods for teaching calculation to primary school students. It covers teaching addition, subtraction, multiplication and division. For each operation, it provides examples of methods like using number lines, partitioning, repeated addition/subtraction, and formal written methods. The goal is to help students develop a strong foundation in understanding numbers and calculations.
This document introduces the Nikhilam Sutra, a method of Vedic mathematics for multiplication. It explains the principles and provides examples of multiplying numbers near and away from multiples of 10 using appropriate bases. The key steps are to make the numbers equal in digits, choose a base, find the differences from the base, add the numbers and differences, and multiply the differences. It also covers cases where the numbers are slightly above or below multiples and proportional methods for numbers with rational relationships. Practice problems are provided to demonstrate applying the sutra.
This document provides instructions and examples for using a math workbook on fractions. It explains how to work through the book step-by-step from less complex to more complex topics like addition, subtraction, multiplication and division of fractions. Examples are provided to demonstrate reducing fractions to a common denominator and performing operations on homogeneous and heterogeneous fractions. Time limits are given for each section to practice problems until they can be completed within the allotted time.
The document provides examples and explanations for solving problems involving patterns and functions. It presents a five-step plan for problem solving: 1) read the problem, 2) plan how to set up and solve, 3) do the work, 4) answer the question, and 5) check the answer. It then works through three examples step-by-step to find patterns in tables and write equations to describe the relationships. The homework assignment is to complete problems 1-20 on page 274.
The document provides information on numerical reasoning concepts including arithmetic progression, geometric progression, formulas, ratio and proportion problems, alligation, and mixture problems. It includes 15 multi-step word problems covering these topics and their step-by-step solutions. The problems demonstrate how to set up and solve ratios, proportions, alligation and mixture scenarios to find unknown values.
This document outlines a three day lesson plan on solving equations using properties of equality. The objectives are to solve linear equations using addition, subtraction, division, and multiplication properties. Students will learn to solve equations of the form x + b = c, ax = c, and ax + b = c. The lessons include step-by-step examples and word problems. Students will practice solving equations on their own. Additional online resources are provided for extra practice.
The document contains 31 multi-step math word problems with solutions. The problems cover a range of topics including percentages, ratios, averages, probability, geometry, and more. The level of difficulty ranges from relatively simple to more complex.
This document contains 6 problems and their solutions from a Regional Mathematical Olympiad competition in 2010. Problem 1 involves proving that the area of one triangle is the geometric mean of the areas of two other triangles in a hexagon with concurrent diagonals. Problem 2 proves that if three quadratic polynomials have a common root, then their coefficients must be equal. Problem 3 counts the number of 4-digit numbers divisible by 4 but not 8 having non-zero digits. Problem 4 finds the smallest positive integers whose reciprocals satisfy certain relationships. Problem 5 proves that the reflection of a point in a line lies on a side of a triangle. Problem 6 determines the number of values of n where an is greater than an+1 for a defined sequence
This document outlines lessons from a mathematics textbook on multiplying multi-digit numbers by two-digit numbers. It includes 7 lessons that cover multiplying by tens and hundreds, estimating products, using different problem-solving strategies like acting it out, standard algorithms for multiplying two-digit and three-digit numbers by two-digit numbers, and choosing the best strategy for a given problem. Examples and practice problems are provided for each lesson.
This document discusses different number systems such as binary, decimal, hexadecimal, and octal. It provides details on how to convert between these number systems using techniques like multiplying each bit by its place value. Examples are given for converting between the different bases to illustrate concepts like binary addition, multiplication, and representing fractions.
This document provides methods, tips, and tricks for multiplying two numbers that start with the same digit using a reference point method. It demonstrates how to solve multiplication problems by determining how much each number is from a chosen reference point, usually a multiple of 10. It shows examples of multiplying numbers using different reference points from 10 to 100 and explains how to choose an appropriate reference point based on the numbers. The key steps are to find the difference of each number from the reference point, multiply those differences and add it to the product of multiplying the numbers' sums from the reference point.
This document contains figures and images related to the digestive system. It includes over 50 labeled figures showing the organs and structures of the digestive system, including the mouth, esophagus, stomach, small intestine, large intestine, liver, pancreas and gallbladder. The figures show the locations and relationships between these organs, as well as microscopic views of tissues and diagrams of metabolic pathways.
E-learning in Medical Education and Blended Learning Approach discusses the history and applications of e-learning in medical education. It begins by explaining how education has shifted from teacher-centered to learner-centered and how e-learning can improve learning outcomes. E-learning uses technology like the internet to provide educational resources beyond the classroom. While e-learning has advantages, traditional learning still has benefits. The document then reviews the history of using computers in medical education from the 1960s onward. It describes different modes of e-learning delivery and applications in medical education today.
Brock biology-of-microorganisms-(13th-edition)Shahab Pour
This document provides an overview of microbiology. It begins with an introduction to microbiology as a science and discusses the historical discoveries that contributed to the field. It then describes techniques used to study microbes like microscopy. It explores the diversity of microbial life including bacteria, archaea, and microbial eukaryotes. The document also examines microbial cell structure, metabolism, growth, genetics, and genomics. It discusses the roles of microbes in areas like photosynthesis, biogeochemical cycles, and commercial applications. Finally, it considers microbial evolution and systematics.
Hamsters, gerbils, and ferrets are common small mammal pets. Hamsters have cheek pouches and flank glands, and require exercise wheels in their cages. Gerbils are adapted to desert environments, have a scent gland on their stomach, and are generally quiet. Ferrets are carnivores descended from European polecats, and require protein-rich commercially available diets or high-quality cat food. Both hamsters and gerbils can be handled gently by lifting behind the front or rear quarters, while ferrets are best lifted by the nape of the neck or with one hand around the body.
Edexcel as chemistry tag 2nd ed (Book answers and teachers guide)Samith Senadeera
This document is a teacher guide for the Edexcel AS Chemistry textbook. It provides answers to questions at the end of each chapter in the textbook to help students understand the concepts. For each answer, additional examiner notes are included to explain the reasoning and common mistakes to avoid. The guide also contains model answers for practice unit tests with marking points indicated. It is designed to help both teachers and students with the Edexcel AS Chemistry curriculum.
The document provides an overview of an introductory chapter on computers and C++ programming. It discusses computer systems and components, including hardware, memory, processors, and software. It then covers programming concepts like algorithms, problem solving approaches, and the C++ programming language. Specific topics include data representation in memory, compilers, object-oriented programming, and the software development lifecycle. Sample C++ code is presented and explained to demonstrate basic program structure and I/O.
This document provides an overview and instructions for an eBook format called ePUB. It discusses that support for ePUB features varies across devices. It recommends customizing settings like font, size, and layout. It notes that some eBooks include code examples that are best viewed in single column landscape mode. Images are included to mimic the print version code display. Clicking links will show the print-fidelity code image. The back button returns to the previous page.
The document is the solutions manual for a fluid mechanics textbook. It contains solutions to example problems from Chapter 1 which covers basic concepts in fluid mechanics, including definitions of different types of flows, fluid properties, forces, and units. The solutions provide concise explanations and calculations for each question with the relevant equations, assumptions, analysis, and discussions of the results.
The document describes steps for solving applied problems using linear equations:
1. Read the problem and identify what is given and what needs to be found.
2. Assign a variable to represent the unknown and write an equation relating the variable to the information given.
3. Solve the equation to find the value of the variable, which represents the answer.
It then provides examples of applying these steps to problems involving unknown numbers, sums of quantities, supplementary and complementary angles, and consecutive integers.
This document discusses linear equations in one variable. It defines linear equations as those involving single variables with the highest power being 1. It presents rules for solving linear equations, including adding, subtracting, multiplying, or dividing the same quantity to both sides. Transposition as a method is explained, where terms change signs when shifted between sides of an equation. Examples of solving linear equations are provided. The document also discusses applying linear equations to word problems by setting up the equation based on the problem and solving for the unknown variable. Several examples of solving word problems involving linear equations are presented.
The document discusses the binomial theorem, which provides a method for expanding binomial expressions of the form (x + y)^n. It explains that each term of the expansion is of the form x^i y^(n-i), with coefficients given by the binomial coefficients. Pascal's triangle is introduced as a way to determine the coefficients. Examples are provided to demonstrate expanding binomial expressions and using the binomial coefficients and theorem.
The document provides steps and examples for solving various types of word problems in algebra, including number, mixture, rate/time/distance, work, coin, and geometric problems. It also covers solving quadratic equations using methods like the square root property, completing the square, quadratic formula, factoring, and using the discriminant. Finally, it discusses linear inequalities, including properties related to addition, multiplication, division, and subtraction of inequalities.
1. The document discusses solving trigonometric equations and finding their general solutions. It provides examples of solving equations using inverse trig functions, factoring, and substitution.
2. General solutions to trig equations involve adding integer multiples of the period (2π or 180°) to the solutions to account for all possibilities in the entire domain.
3. Examples show solving equations like cosx = 0.456 by taking the inverse cosine and factoring equations like 3tan^2x + 4tanx + 1 = 0 to find specific solutions and the general form.
The document discusses properties of equalities and inequalities as well as how to solve linear equations and inequalities with one variable. It introduces properties of equality like the addition, subtraction, multiplication, and division properties. It also covers properties of operations like the commutative, associative, and distributive properties. Properties of inequality are presented along with how to use properties to solve equations and inequalities with one variable by manipulating and isolating the variable. Examples are provided to demonstrate solving linear equations and graphing solutions to linear inequalities on a number line.
The document discusses properties of equalities and inequalities as well as how to solve linear equations and inequalities with one variable. It introduces properties of equality like the addition, subtraction, multiplication, and division properties. It also covers properties of operations like commutative, associative, and distributive properties. Properties of inequality are presented along with how to use properties to solve equations and inequalities with one variable by adding, subtracting, or isolating the variable. Examples are provided to demonstrate solving linear equations and graphing solutions to linear inequalities on a number line.
This document provides an overview of linear systems and their application to economic models. It contains the following key points:
1) Wassily Leontief used systems of linear equations to model economies and make predictions, winning a Nobel Prize in Economics.
2) An example economic model is given with two industries (goods, services) and equations showing their internal demands.
3) The example forms equations to model the total demand and supply of each industry, and solves the system to find the dollar values each industry must produce.
4) Methods for solving systems of linear equations are discussed, including substitution and elimination approaches. The concepts of consistency and uniqueness of solutions are also introduced.
This document defines key mathematical terms related to subsets, properties, ratios, proportions, percentages, and interest. It includes:
- Definitions of natural numbers, whole numbers, and integers as subsets of numbers
- Explanations of closure, commutative, associative, and identity properties
- How to calculate ratios, rates, and use proportions
- Conversions between decimals, fractions, and percentages
- A formula for calculating simple interest (I=PRT) and an example using it
The document provides concise explanations and examples of fundamental mathematical concepts.
The document provides a review of multiplication and division concepts including:
1) Examples of relating multiplication and division such as reversing number sentences.
2) Practice problems involving variables such as determining values of variables in equations.
3) A word problem involving dividing a total number of students among classes of equal size.
Real numbers include rational numbers like fractions and irrational numbers like square roots. Real numbers are represented by the symbol R. They consist of natural numbers, whole numbers, integers, rational numbers and irrational numbers. [/SUMMARY]
Project in math BY:Samuel Vasquez Baliasamuel balia
Real numbers include rational numbers like fractions as well as irrational numbers like the square root of 2. Real numbers are represented by the symbol R and include natural numbers, whole numbers, integers, rational numbers, and irrational numbers. Rational numbers can be written as fractions with integers as the numerator and non-zero denominator, while irrational numbers cannot be expressed as fractions.
Linear Equation Word Problem Complete.pptJeneferburdas
This document provides guidance on translating word problems into mathematical expressions and equations. It includes examples of common phrases expressed mathematically and steps for solving applied problems. Key translation topics covered are addition, subtraction, multiplication, division, equality, and distinguishing expressions from equations. Example word problems presented at the end demonstrate solving for unknown values in various contexts like geometry, mixtures, percentages, and investments.
Quadratic equations are polynomial equations of the second degree that take the general form of ax2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. They can be solved using methods like factoring, completing the square, or the quadratic formula. The discriminant, b2 - 4ac, determines whether the roots are real numbers, a repeated real root, or complex numbers.
1) The document defines terms, constants, coefficients, and discusses how to solve one-step and two-step equations. It provides examples of solving equations involving addition, subtraction, multiplication and division.
2) Sample one-step equations are provided along with the steps to solve each type. Equations involving addition, subtraction, multiplication and division are worked through as examples.
3) A two-step practice problem is given along with the answers. Solving two-step equations is introduced.
This document provides examples of how to translate common English language expressions about mathematical operations like addition, subtraction, multiplication, and division into algebraic expressions. It explains key words that indicate operations and equations. Several examples of word problems are presented along with the steps to solve them, which include assigning variables, writing equations, solving equations, and checking answers. Common types of word problems discussed include geometry, percentages, investments, mixtures, and more.
This document provides instruction on solving quadratic equations by factoring. It begins by explaining what factoring is and how to factor trinomials of the form x^2 + bx + c. Examples are given of factoring different types of trinomials, including those where b is negative or c is negative. The document then explains how to use the zero product property to solve quadratic equations after factoring them into binomial form. Several examples of solving equations by factoring are shown. Finally, applications of factoring quadratic equations to area word problems are presented.
The document defines linear programming as a branch of mathematics used to find the optimal solution to problems with constraints. It provides examples of using linear programming to maximize profit or minimize costs in organizations. It also introduces drawing linear inequalities and solving simultaneous inequalities. The steps to formulate a linear programming problem are identified as defining variables and objectives, translating constraints, finding feasible solutions, and evaluating objectives to find optimal solutions.
The document provides examples and explanations for translating word problems and phrases into algebraic expressions. It gives examples such as "18 less than a number" being translated to "x - 18" and "the product of a number and 5" being "5n". It also provides word problems like writing an expression for the total cost of admission plus rides at a county fair. The document teaches learners how to identify keywords that indicate mathematical operations when translating word phrases into algebraic notation.
Similar to 8.6+applications+and+problem+solving (20)