The dev project
Download as PPTX, PDF0 likes318 views
This document contains 4 math problems with step-by-step solutions: 1) factoring and graphing a polynomial function, finding vertical and horizontal asymptotes, x-intercepts, and holes; 2) finding the maximum area of a garden within a fence of a given length; 3) fully distributing a polynomial; 4) factoring a quadratic equation.
1 of 19
Download to read offline
![To find the maximum value use the formula -b/2a to
find the x of the vertex
-200/-4= 50
Plug x into the equation to get the maximum
-2(50)族+200(50)=5000
D:(-,)
No number x cannot equal
R:(-,5000]
No minimum value and already found the
maximum](https://image.slidesharecdn.com/thedevproject-131118221603-phpapp02/85/The-dev-project-10-320.jpg)
Recommended
Maths formula by viveksingh698@gmail.com
Maths formula by viveksingh698@gmail.comvivek698
油
The document outlines 9 multiplication shortcuts or tricks using properties of numbers. These include multiplying numbers by 11 by adding the digits, squaring numbers ending in 9 by placing 9s and appending other digits, squaring numbers ending in 5 by omitting the 5 and multiplying the remaining number by the next higher number and appending 25, and multiplying numbers where the ones digits sum to 10 by multiplying the tens and ones places separately and placing the products successively.Dividing polynomials
Dividing polynomialssalvie alvaro
油
This document provides instructions on dividing polynomials using long division and synthetic division. It explains that long division can be used to divide polynomials just as it is used to divide numbers. An example of long division of polynomials is shown. Synthetic division is introduced as an alternative method that is used when the coefficient of the highest term is 1. Examples of both long division and synthetic division of polynomials are provided and step-by-step workings are shown. Some notes are included that synthetic division should be used when the coefficient is 1 and that testing for zero remainder determines if a binomial is a factor.Synthetic Division
Synthetic Divisionscnbmitchell
油
The document provides an explanation and examples of using synthetic division to divide polynomials. Synthetic division allows dividing a polynomial by a divisor of the form (x - k). The process involves writing the coefficients of the dividend in descending order and placing k below. Then, successive multiplication and addition steps provide the coefficients of the quotient polynomial and remainder. Two examples are worked through to demonstrate synthetic division for (2x^3 - 7x^2 - 8x + 16) / (x - 4) and (5x^3 + x^2 - 7) / (x + 1).Finding zeros of a quadratic function
Finding zeros of a quadratic functionAaron James Lico
油
The document provides examples of finding the zeros of quadratic functions by factorizing and setting each factor equal to 0. It then lists 5 additional quadratic functions and assigns the reader to find their zeros.Solving Quadratic Equations
Solving Quadratic Equationsdowne1mf
油
This document discusses how to solve quadratic equations by factoring, using the quadratic formula, and determining the number and type of roots using the discriminant. Key steps include factoring the equation if possible to set each factor equal to 0 and solve, plugging the coefficients a, b, and c into the quadratic formula if factoring is not possible, and using the discriminant b^2 - 4ac to determine the number and type of roots. Examples are provided to demonstrate each method.Dividing polynomials
Dividing polynomials掘糸顎界温界庄坦稼
油
1) The document explains various methods for dividing and factoring polynomials, including: dividing polynomials using long division; using Ruffini's rule to divide polynomials; applying the remainder theorem and factor theorem; and factoring polynomials through finding common factors, using identities, solving quadratic equations, and finding polynomial roots.
2) Specific factorization methods covered are removing common factors, using identities like a^2 - b^2, factoring quadratic trinomials, using the remainder theorem and Ruffini's rule to find factors for polynomials of degree greater than two, and identifying irreducible polynomials.
3) Additional algebraic identities explained are for cubing binomials like (a 賊 b)^3 and taking the square of trinomialDEV Project
DEV ProjectLuciusAtkinson
油
1. The document provides instructions for solving 4 math challenges involving quadratic equations, factoring polynomials, combining functions, and finding the domain and range of a radical function.
2. The challenges include completing the square to solve a quadratic equation, factoring a polynomial using grouping, combining two functions by substituting one function into the other, and simplifying a radical function before determining its domain.
3. The steps provided include factoring expressions, simplifying functions, determining individual domains before combining, and using logical rules to derive the combined domain of a radical function.06 - 20 Jan - Divide and Conquer
06 - 20 Jan - Divide and ConquerNeeldhara Misra
油
The document discusses algorithms for finding the closest pair of points from a set of n points given as (x,y) coordinates. It presents three approaches: computing all pairwise distances which is O(n^2); sorting the points and comparing adjacent pairs which is the time to sort plus O(n); and divide and conquer which splits the points, solves subproblems recursively, and merges in O(n log n) time.Polynomials Grade 10
Polynomials Grade 10ingroy
油
This document provides instructions on how to multiply, divide, and factor polynomials. It discusses:
1) Multiplying polynomials by distributing terms and using FOIL for binomials.
2) Dividing polynomials using long division.
3) Factoring polynomials using grouping, finding two numbers whose product is the constant and sum is the coefficient, and recognizing difference of squares.The Quadratic Function Derived From Zeros of the Equation (SNSD Theme)
The Quadratic Function Derived From Zeros of the Equation (SNSD Theme)SNSDTaeyeon
油
Christian Jett Morales presents on the quadratic function derived from zeros of the function. The presentation includes:
- An introduction to deriving the quadratic function f(x) = ax^2 + bx + c from the zeros r1 and r2 of the function.
- Examples of deriving quadratic functions when given different zeros, such as 3/2 and -1, or 2+3 and 2-3.
- An activity for participants to practice deriving quadratic functions.
- A dedication to God and the presenter's teacher for the opportunity to create the presentation.Polynomial Function and Synthetic Division
Polynomial Function and Synthetic DivisionElceed
油
This presentation explains the basic information about Polynomial Function and Synthetic Division. Examples were given about easy ways to divide polynomial function using synthetic division. It also contains the steps on how to perform the division method of polynomial functions. Polynomial Division
Polynomial Divisionjulietemelda
油
This document discusses various techniques for solving polynomial expressions using long division. It provides examples of:
1) Dividing polynomials with common variables and subtracting exponents
2) Simplifying polynomial expressions using factoring or long division
3) Dividing polynomials where one term is missing by treating the missing term as zero
4) Using long division to solve word problems involving areas of shapesAlgebra ii honors study guide
Algebra ii honors study guidemorrobea
油
1. The document contains 50 math problems involving ordering numbers, evaluating expressions, solving equations and inequalities, graphing functions and relations, and other topics.
2. For each problem, the key steps are shown and the solution is provided in brackets at the end in reference to the problem number.
3. The document serves as an expert summary by providing concise 3-sentence or less solutions for each of the 50 math problems.Expresii algebrice-rezolvate
Expresii algebrice-rezolvateGherghescu Gabriel
油
The document contains solutions to several algebra problems involving expressions. For the first problem, the values of x where the expression E(x) is defined are found to be between -3, 3, -2/2, and 5 excluding -3, 3, and 5. The values of a where E(a) is an integer are {-11, -7, -5, -4, -2, -1, 1}. The solution set to the inequality is {0, 1, 2, 4}. For the second problem, the values of x where the expression is undefined are -1, 3, and 10. It is shown that the expression equals 5-x/(x-3) when evaluated atBinomial theorem for any index
Binomial theorem for any indexindu psthakur
油
The document summarizes the binomial theorem and properties of binomial coefficients. It provides:
1) The binomial theorem expresses the expansion of (a + b)n as a sum of terms involving binomial coefficients for any positive integer n.
2) Important properties of binomial coefficients are discussed, such as their relationship to factorials and the symmetry of coefficients.
3) Examples are given of using the binomial theorem to find coefficients and solve problems involving divisibility and series of binomial coefficients.basic concepts of Functions
basic concepts of FunctionsTarun Gehlot
油
The document defines domain and range for functions. The domain is the set of all x-coordinates in a set of ordered pairs, while the range is the set of all y-coordinates. The document provides examples of finding the domain and range for finite sets of ordered pairs defined by lists, infinite sets defined by graphs, and functions defined by algebraic expressions. Practice problems are given along with the solutions for finding the domain and range in different scenarios.The binomial theorem class 11 maths
The binomial theorem class 11 mathsDharmendra Dudi
油
This document discusses binomial expansion, which is the process of expanding expressions with two terms like (x + a) to higher powers without lengthy multiplication. It introduces Pascal's triangle as a way to determine the coefficients in the expanded terms. It then defines the factorial operation and provides a general formula for determining the coefficients of any term when expanding a binomial to a given power.Module 2 polynomial functions
Module 2 polynomial functionsdionesioable
油
This module discusses methods for finding the zeros of polynomial functions of degree greater than 2, including: factor theorem, factoring, synthetic division, and depressed equations. It introduces the number of roots theorem, which states that a polynomial of degree n has n roots. It also discusses determining the rational zeros of a polynomial using the rational roots theorem and factor theorem. Examples are provided to illustrate these concepts and methods.Binomial
BinomialDEDESUDJADI
油
The document provides an introduction to the binomial theorem. It begins by discussing binomial coefficients through the Pascal's triangle. It then derives an explicit formula for binomial coefficients using factorials. Finally, it states the binomial theorem and provides examples of using it to expand algebraic expressions and estimate numerical values.Series expansion of exponential and logarithmic functions
Series expansion of exponential and logarithmic functionsindu psthakur
油
The document summarizes series expansions of exponential and logarithmic functions. It provides the series expansions of ex, e-x, ex+c, and logarithmic functions like log(1+x). It discusses the properties of the natural logarithm and exponential functions. Examples are given of finding sums of various exponential and logarithmic series. Questions with hints are also provided about proving properties related to logarithmic and exponential series.Solving quadratics with square roots
Solving quadratics with square rootsswartzje
油
This document provides information on solving quadratic equations by finding square roots. It discusses how to solve equations by taking the square root of each side, and that the square root of a positive number has two answers: one positive and one negative. Properties of square roots are presented. Examples are provided on simplifying expressions involving square roots using the product and quotient properties. The document also covers rationalizing the denominator to remove radicals in fractions and solving quadratic equations by taking square roots and isolating the variable. Methods for solving word problems involving falling objects using the quadratic formula are demonstrated.Integration
Integrationtimschmitz
油
This document provides an overview of integration and how it relates to calculating areas under curves and between functions. Some key points covered include:
- Integration uses the concept of reverse differentiation to calculate the area under a function.
- The area under a function between two x-values a and b is calculated as the definite integral from a to b.
- Integrals may be used to find the area between two curves by subtracting their integrals between the intersection points.
- Integrals can represent either positive or negative areas depending on whether the area is above or below the x-axis.QUADRATIC EQUATIONS
QUADRATIC EQUATIONShiratufail
油
This document discusses four methods for solving quadratic equations: factorization, completing the square, using a formula, and using graphs. It provides an example of solving the equation 2x^2 - 10x + 12 + x^2 + 6x = -9 by factorizing into (x - 3)(x - 1) = 0, finding that the solutions are x = 3 or x = 1.Higher Maths 2.1.2 - Quadratic Functions
Higher Maths 2.1.2 - Quadratic Functionstimschmitz
油
This document provides an overview of quadratic functions including:
- The graph of a quadratic function is a parabola.
- The general form of a quadratic function is f(x) = ax2 + bx + c.
- Key aspects that must be identified before sketching the graph are the nature of the turning point, y-intercept, x-intercepts, and axis of symmetry.
- Quadratic equations can be solved by factorizing, completing the square, or using the quadratic formula. The discriminant determines the nature of the roots.1631 the binomial theorem
1631 the binomial theoremDr Fereidoun Dejahang
油
This document discusses binomial expansion, which is a method for expanding binomials like (x + a)^n without lengthy multiplication. It introduces key concepts like Pascal's triangle for finding coefficients and the binomial theorem for determining the general pattern of terms in the expansion. Examples are worked through to demonstrate expanding specific binomials like (2x - 3y)^6 according to this method.Integration
IntegrationSharingIsCaring1000
油
This document provides information about integration in higher mathematics. It begins with an overview of integration as the opposite of differentiation. It then discusses using antidifferentiation to find integrals by reversing the power rule for differentiation. Several examples are provided to illustrate integrating polynomials. The document also discusses using integrals to find the area under a curve or between two curves. It provides examples of calculating areas bounded by graphs and the x-axis. Finally, it presents some exam-style integration questions for practice.
12 12-16Manuel Tellechea
油
El documento analiza el precio del petr坦leo despu辿s de que OPEC y pa鱈ses no miembros llegaran a un acuerdo. El precio salt坦 de $39 a $54, y las medias m坦viles de corto plazo cruzaron al alza. La pr坦xima resistencia est叩 cerca de $62, aunque en el diario hay un patr坦n de reversi坦n bajista. La tendencia general sigue siendo alcista.
Tata nama neoplasma
Tata nama neoplasmahelmy lisik miko
油
cara membedakan tumor jinak (benigna) atau tumor ganas (maligna) . #miko
El tiempo libre de nuestros hijosOrientateca de OrienTapas
油
Recopilaci坦n de art鱈culos sobre el uso del tiempo libre, ocio y deporte, por la orientadora Ana Torres Jack. Difundidos en prensa en los 炭ltimos a単os y en la Orientateca de OrienTapas.More Related Content
What's hot (18)
Polynomials Grade 10
Polynomials Grade 10ingroy
油
This document provides instructions on how to multiply, divide, and factor polynomials. It discusses:
1) Multiplying polynomials by distributing terms and using FOIL for binomials.
2) Dividing polynomials using long division.
3) Factoring polynomials using grouping, finding two numbers whose product is the constant and sum is the coefficient, and recognizing difference of squares.The Quadratic Function Derived From Zeros of the Equation (SNSD Theme)
The Quadratic Function Derived From Zeros of the Equation (SNSD Theme)SNSDTaeyeon
油
Christian Jett Morales presents on the quadratic function derived from zeros of the function. The presentation includes:
- An introduction to deriving the quadratic function f(x) = ax^2 + bx + c from the zeros r1 and r2 of the function.
- Examples of deriving quadratic functions when given different zeros, such as 3/2 and -1, or 2+3 and 2-3.
- An activity for participants to practice deriving quadratic functions.
- A dedication to God and the presenter's teacher for the opportunity to create the presentation.Polynomial Function and Synthetic Division
Polynomial Function and Synthetic DivisionElceed
油
This presentation explains the basic information about Polynomial Function and Synthetic Division. Examples were given about easy ways to divide polynomial function using synthetic division. It also contains the steps on how to perform the division method of polynomial functions. Polynomial Division
Polynomial Divisionjulietemelda
油
This document discusses various techniques for solving polynomial expressions using long division. It provides examples of:
1) Dividing polynomials with common variables and subtracting exponents
2) Simplifying polynomial expressions using factoring or long division
3) Dividing polynomials where one term is missing by treating the missing term as zero
4) Using long division to solve word problems involving areas of shapesAlgebra ii honors study guide
Algebra ii honors study guidemorrobea
油
1. The document contains 50 math problems involving ordering numbers, evaluating expressions, solving equations and inequalities, graphing functions and relations, and other topics.
2. For each problem, the key steps are shown and the solution is provided in brackets at the end in reference to the problem number.
3. The document serves as an expert summary by providing concise 3-sentence or less solutions for each of the 50 math problems.Expresii algebrice-rezolvate
Expresii algebrice-rezolvateGherghescu Gabriel
油
The document contains solutions to several algebra problems involving expressions. For the first problem, the values of x where the expression E(x) is defined are found to be between -3, 3, -2/2, and 5 excluding -3, 3, and 5. The values of a where E(a) is an integer are {-11, -7, -5, -4, -2, -1, 1}. The solution set to the inequality is {0, 1, 2, 4}. For the second problem, the values of x where the expression is undefined are -1, 3, and 10. It is shown that the expression equals 5-x/(x-3) when evaluated atBinomial theorem for any index
Binomial theorem for any indexindu psthakur
油
The document summarizes the binomial theorem and properties of binomial coefficients. It provides:
1) The binomial theorem expresses the expansion of (a + b)n as a sum of terms involving binomial coefficients for any positive integer n.
2) Important properties of binomial coefficients are discussed, such as their relationship to factorials and the symmetry of coefficients.
3) Examples are given of using the binomial theorem to find coefficients and solve problems involving divisibility and series of binomial coefficients.basic concepts of Functions
basic concepts of FunctionsTarun Gehlot
油
The document defines domain and range for functions. The domain is the set of all x-coordinates in a set of ordered pairs, while the range is the set of all y-coordinates. The document provides examples of finding the domain and range for finite sets of ordered pairs defined by lists, infinite sets defined by graphs, and functions defined by algebraic expressions. Practice problems are given along with the solutions for finding the domain and range in different scenarios.The binomial theorem class 11 maths
The binomial theorem class 11 mathsDharmendra Dudi
油
This document discusses binomial expansion, which is the process of expanding expressions with two terms like (x + a) to higher powers without lengthy multiplication. It introduces Pascal's triangle as a way to determine the coefficients in the expanded terms. It then defines the factorial operation and provides a general formula for determining the coefficients of any term when expanding a binomial to a given power.Module 2 polynomial functions
Module 2 polynomial functionsdionesioable
油
This module discusses methods for finding the zeros of polynomial functions of degree greater than 2, including: factor theorem, factoring, synthetic division, and depressed equations. It introduces the number of roots theorem, which states that a polynomial of degree n has n roots. It also discusses determining the rational zeros of a polynomial using the rational roots theorem and factor theorem. Examples are provided to illustrate these concepts and methods.Binomial
BinomialDEDESUDJADI
油
The document provides an introduction to the binomial theorem. It begins by discussing binomial coefficients through the Pascal's triangle. It then derives an explicit formula for binomial coefficients using factorials. Finally, it states the binomial theorem and provides examples of using it to expand algebraic expressions and estimate numerical values.Series expansion of exponential and logarithmic functions
Series expansion of exponential and logarithmic functionsindu psthakur
油
The document summarizes series expansions of exponential and logarithmic functions. It provides the series expansions of ex, e-x, ex+c, and logarithmic functions like log(1+x). It discusses the properties of the natural logarithm and exponential functions. Examples are given of finding sums of various exponential and logarithmic series. Questions with hints are also provided about proving properties related to logarithmic and exponential series.Solving quadratics with square roots
Solving quadratics with square rootsswartzje
油
This document provides information on solving quadratic equations by finding square roots. It discusses how to solve equations by taking the square root of each side, and that the square root of a positive number has two answers: one positive and one negative. Properties of square roots are presented. Examples are provided on simplifying expressions involving square roots using the product and quotient properties. The document also covers rationalizing the denominator to remove radicals in fractions and solving quadratic equations by taking square roots and isolating the variable. Methods for solving word problems involving falling objects using the quadratic formula are demonstrated.Integration
Integrationtimschmitz
油
This document provides an overview of integration and how it relates to calculating areas under curves and between functions. Some key points covered include:
- Integration uses the concept of reverse differentiation to calculate the area under a function.
- The area under a function between two x-values a and b is calculated as the definite integral from a to b.
- Integrals may be used to find the area between two curves by subtracting their integrals between the intersection points.
- Integrals can represent either positive or negative areas depending on whether the area is above or below the x-axis.QUADRATIC EQUATIONS
QUADRATIC EQUATIONShiratufail
油
This document discusses four methods for solving quadratic equations: factorization, completing the square, using a formula, and using graphs. It provides an example of solving the equation 2x^2 - 10x + 12 + x^2 + 6x = -9 by factorizing into (x - 3)(x - 1) = 0, finding that the solutions are x = 3 or x = 1.Higher Maths 2.1.2 - Quadratic Functions
Higher Maths 2.1.2 - Quadratic Functionstimschmitz
油
This document provides an overview of quadratic functions including:
- The graph of a quadratic function is a parabola.
- The general form of a quadratic function is f(x) = ax2 + bx + c.
- Key aspects that must be identified before sketching the graph are the nature of the turning point, y-intercept, x-intercepts, and axis of symmetry.
- Quadratic equations can be solved by factorizing, completing the square, or using the quadratic formula. The discriminant determines the nature of the roots.1631 the binomial theorem
1631 the binomial theoremDr Fereidoun Dejahang
油
This document discusses binomial expansion, which is a method for expanding binomials like (x + a)^n without lengthy multiplication. It introduces key concepts like Pascal's triangle for finding coefficients and the binomial theorem for determining the general pattern of terms in the expansion. Examples are worked through to demonstrate expanding specific binomials like (2x - 3y)^6 according to this method.Integration
IntegrationSharingIsCaring1000
油
This document provides information about integration in higher mathematics. It begins with an overview of integration as the opposite of differentiation. It then discusses using antidifferentiation to find integrals by reversing the power rule for differentiation. Several examples are provided to illustrate integrating polynomials. The document also discusses using integrals to find the area under a curve or between two curves. It provides examples of calculating areas bounded by graphs and the x-axis. Finally, it presents some exam-style integration questions for practice.Viewers also liked (10)
12 12-16Manuel Tellechea
油
El documento analiza el precio del petr坦leo despu辿s de que OPEC y pa鱈ses no miembros llegaran a un acuerdo. El precio salt坦 de $39 a $54, y las medias m坦viles de corto plazo cruzaron al alza. La pr坦xima resistencia est叩 cerca de $62, aunque en el diario hay un patr坦n de reversi坦n bajista. La tendencia general sigue siendo alcista.Tata nama neoplasma
Tata nama neoplasmahelmy lisik miko
油
cara membedakan tumor jinak (benigna) atau tumor ganas (maligna) . #mikoEl tiempo libre de nuestros hijosOrientateca de OrienTapas
油
Recopilaci坦n de art鱈culos sobre el uso del tiempo libre, ocio y deporte, por la orientadora Ana Torres Jack. Difundidos en prensa en los 炭ltimos a単os y en la Orientateca de OrienTapas.Kul 3. Morfologi jejas sel
Kul 3. Morfologi jejas selRobby Candra Purnama
油
Membahas bagaimana morfologi (bentuk) jejas sel dan mekanisme terbentuk jejas selOpenStack Journey in Tieto Elastic Cloud
OpenStack Journey in Tieto Elastic CloudJakub Pavlik
油
This document summarizes Tieto's journey with OpenStack, beginning with initial proof of concept deployments using installer-based distributions like PackStack and Fuel. Tieto found these distributions difficult to maintain and customize. They later implemented tcp cloud using Mirantis OpenStack and OpenContrail, which provided better automation and configuration management. While this worked well initially, managing over 30 management VMs became challenging. Tieto is now moving to Mirantis Cloud Platform, which runs OpenStack services on Kubernetes for improved flexibility, lifecycle management, and reduced management overhead.
Folletos de Orientaci坦n para familias y tutoresOrientateca de OrienTapas
油
Presentaci坦n de Mariola Rodr鱈guez. Finalista de las Experiencias #AulaDelOrientador 2015: http://orientapas.blogspot.com.es/p/experiencias-auladelorientador-2015.htmlDifficult interactions - Part 2
Difficult interactions - Part 2LifeatLeggett
油
This is part two of two installments on how to manage difficult interactions. It focuses on how to neutralize misunderstandings before they turn into conflict as well as what to do when the difficult interaction is unavoidable.English holidays
English holidaysOlexandr Rybalko School of Klavdievo
油
The document contains phonetic drills, rhymes, vocabulary words and phrases related to English holidays such as St. Patrick's Day and Easter. It discusses traditions like wearing shamrocks on St. Patrick's Day and features the rhyme "Pussy Cat" along with vocabulary, listening exercises and grammar lessons focused on the past simple tense using irregular verbs. The document provides English language learning content for students through poems, drills and exercises centered around seasonal celebrations.
Similar to The dev project (20)
Advance algebra
Advance algebralyra matalubos
油
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.Quadratic equations
Quadratic equationsMervin Dayrit
油
This document discusses quadratic equations and methods for solving them. It begins by defining quadratic equations as second degree polynomial equations of the form ax^2 + bx + c = 0, where a is not equal to 0. It then presents several methods for finding the roots or solutions of quadratic equations: factoring, completing the square, and using the quadratic formula. Examples are provided to illustrate each method. The document also discusses graphing quadratic functions and key features of parabolas such as vertex, axis of symmetry, and direction of opening.Mc ty-cubicequations-2009-1
Mc ty-cubicequations-2009-1Harsh Arora
油
1) Cubic equations can have one real root or three real roots. They always have at least one real root.
2) Cubic equations can be solved by using synthetic division to reduce them to a quadratic equation if one root is known, or by spotting factors.
3) Graphs of cubic equations cross the x-axis at least once, showing they have at least one real root, and can help locate approximate solutions.Mc ty-cubicequations-2009-1
Mc ty-cubicequations-2009-1Brit4
油
1) Cubic equations can have one real root or three real roots. They always have at least one real root.
2) Cubic equations can be solved by using synthetic division to reduce them to a quadratic equation if one root is known, or by spotting factors.
3) Graphs of cubic equations cross the x-axis at least once, showing they have at least one real root, and can help locate approximate solutions.Humaira quadratic
Humaira quadratictkhan25
油
The document discusses various topics relating to quadratic equations including:
1) Methods for solving quadratic equations such as completing the square, factorization, and the quadratic formula.
2) The steps for using the completing the square method which involves rewriting the quadratic in the form of a perfect square trinomial.
3) Examples are provided to demonstrate how to use completing the square to solve quadratic equations.Quadratic Equations
Quadratic EquationsWenslette Rosique
油
Quadratic equations take the form ax^2 + bx + c = 0. This document discusses four methods for solving quadratic equations: factorizing, completing the square, using the quadratic formula, and graphing. It provides examples of solving quadratic equations with each method and emphasizes that practice is needed to master the techniques.D.E.V. Project- Precalculus Functions
D.E.V. Project- Precalculus Functionshilschroeds19
油
Final project for Precalc. Walk through of 4 different problems from different units that we studied.Quadraticfunctionpresentation 100127142417-phpapp02
Quadraticfunctionpresentation 100127142417-phpapp02Vine Gonzales
油
The document defines quadratic functions and discusses their various forms, including general, vertex, and factored forms. It also covers solving quadratic equations using methods like the quadratic formula, factoring, and completing the square. Additionally, it discusses key features of quadratic graphs like x-intercepts, y-intercepts, the vertex, and concavity. Examples are provided to illustrate finding these features and graphing parabolas.Jackson d.e.v.
Jackson d.e.v.Dougfield32
油
The document provides step-by-step instructions for factoring polynomials, finding inverse functions, simplifying rational expressions, and graphing rational functions. It includes examples of each type of problem worked out in detail from beginning to end. The examples range from relatively simple to more complex in order to demonstrate a variety of situations that may occur.Developing Expert Voices
Developing Expert Voicessuzanne
油
The student reflects on completing a math project for their calculus course as a way to study for an upcoming exam. They acknowledge that they procrastinated significantly but were able to cover a broad range of calculus concepts through multi-step word problems selected from different units. While the assignment did not dramatically increase their knowledge, it helped reinforce some details and connections between topics. The student resolves to select deadlines more wisely and stop procrastinating for future projects.Quadratic Function Presentation
Quadratic Function PresentationRyanWatt
油
The document defines and explains key concepts regarding quadratic functions including:
- The three common forms of quadratic functions: general, vertex, and factored form
- How to find the x-intercepts, y-intercept, and vertex of a quadratic function
- Methods for solving quadratic equations including factoring, completing the square, and the quadratic formula
- How to graph quadratic functions by identifying intercepts and the vertexPerfect Squares
Perfect Squaresguest02fe304
油
The document discusses perfect squares and how to identify, calculate, and factor them. It provides examples of perfect square numbers and binomials. There are three rules for calculating perfect squares: square the first term, multiply both terms and double the product, and square the last term. There are also formulas for addition and subtraction of perfect squares. The document demonstrates how to factor a trinomial back into a perfect square binomial using the reverse of the formulas.Module 4 quadratic functions
Module 4 quadratic functionsdionesioable
油
This module discusses quadratic functions and their application to solving word problems. Students will learn to 1) recall steps to solve word problems, 2) translate problems to symbolic expressions, and 3) apply quadratic equations. Examples provided cover number, geometry, and motion problems. Students must first translate word problems into equations before solving. Key steps include identifying unknowns, writing the equation, solving, and checking solutions. Practice problems are provided to help students apply the concepts.3.1 methods of division t
3.1 methods of division tmath260
油
Using long division, polynomials can be divided into a quotient and remainder. The remainder has a lower degree than the divisor. Long division and synthetic division allow polynomials to be divided and any roots or factors identified. Various examples demonstrate dividing polynomials using long division and synthetic division, writing the division in the form of quotient plus remainder over divisor. Exercises provide additional practice problems dividing polynomials using long division and synthetic division.Ca 1.6
Ca 1.6salamhello
油
This document provides an overview of quadratic equations, including definitions, methods for solving quadratic equations such as factoring, completing the square, and using the quadratic formula, and applications of quadratic equations. Key topics covered include defining linear and quadratic equations, solving quadratics by factoring when possible and using completing the square or the quadratic formula when not factorable, deriving the quadratic formula, interpreting the discriminant, and modeling real-world situations with quadratic equations.Solution 3
Solution 3aldrins
油
The document provides the steps to solve a multi-part calculus problem. It involves finding the derivative of two functions, determining the equations of tangent lines to those functions at given points, finding the intersection points of those tangent lines, and using those intersection points to write the equations of three circles with a radius of 5.Solution 3
Solution 3aldrins
油
The document provides the steps to solve a multi-part calculus problem involving derivatives, tangent lines, and circles. It determines the derivative of two functions f(x) and g(x), finds the equations of the tangent lines at specific x-values, identifies the intersection points of the tangent lines, and uses those intersection points as the centers of three circles with a radius of 5 to write the equations of the circles.Differential Equations Homework Help
Differential Equations Homework HelpMath Homework Solver
油
I am Leonard K. I am a Differential Equations Homework Solver at mathhomeworksolver.com. I hold a Master's in Mathematics From California, USA. I have been helping students with their homework for the past 8 years. I solve homework related to Differential Equations.
Visit mathhomeworksolver.com or email support@mathhomeworksolver.com. You can also call on +1 678 648 4277 for any assistance with Differential Equations Homework. Algebra slideshow
Algebra slideshowAlgebraProject
油
The document provides instructions for graphing and solving various types of quadratic equations. It defines standard form, vertex form, and intercept form of quadratics. It explains how to graph quadratics by finding the vertex and intercepts. Methods covered include factoring, taking square roots, completing the square, and using the quadratic formula. Examples are included to demonstrate each process.Modeling with Quadratics
Modeling with QuadraticsPLeach
油
- The document discusses quadratic functions and their graphs. It explains that the graph of a quadratic function is a parabola, which is a U-shaped curve.
- It describes how to write quadratic functions in standard form and use that form to sketch the graph and find features like the vertex and axis of symmetry.
- Examples are provided to demonstrate how to graph quadratic functions in standard form and how to find the minimum or maximum value of a quadratic function by setting its derivative equal to zero.Recently uploaded (12)
How I Envision Pakistan by the year 2035.pdf
How I Envision Pakistan by the year 2035.pdfheyitsyouhere123
油
Pakistan stands at a crossroads, poised for transformation into a prosperous, knowledge-driven nation by 2035. This essay envisions a future where economic innovation, social empowerment, and sustainable development converge to shape a dynamic and inclusive society. Through strategic investments in education, technology, and governance, Pakistan can harness its youthful potential to drive progress across industries. Key pillars of this vision include fostering a knowledge-based economy, implementing governance reforms for transparency and efficiency, and championing environmental stewardship. By prioritizing renewable energy, digital transformation, and global engagement, Pakistan can emerge as a regional leader, leveraging its cultural heritage and strategic location for economic and diplomatic growth. This essay presents a roadmap for turning challenges into opportunities, ensuring that every citizen benefits from a more equitable, innovative, and sustainable Pakistan by 2035.Mastering-Emotional-Intelligence.Presentation
Mastering-Emotional-Intelligence.PresentationOzias Rondon
油
Emotional intelligence (EQ) is one of the most important skills for personal and professional success in todays world. This comprehensive presentation explores the key elements of EQ: self-awareness, self-regulation, motivation, empathy, and social skills. You'll learn how to recognize and manage your own emotions, communicate more effectively, build stronger relationships, and make better decisionsespecially under pressure. Whether you're a student, leader, entrepreneur, or professional, developing your emotional intelligence can dramatically improve your career, relationships, and overall well-being.
AOP (Aspect-Oriented Programming) spring boot
AOP (Aspect-Oriented Programming) spring bootPLAYAFIFI
油
AOP (Aspect-Oriented Programming) spring boot
Mind-Mapping-How-to-Organize-Your-Ideas-Like-a-Pro
Mind-Mapping-How-to-Organize-Your-Ideas-Like-a-ProOzias Rondon
油
In this presentation, we'll explore the power of mind mapping as a tool to organize your ideas, boost creativity, and improve productivity. Learn how to visually structure your thoughts, plan projects effectively, and make complex information easier to understand. Whether you're tackling personal goals, academic projects, or business strategies, mind mapping will help you see the bigger picture and keep your ideas flowing smoothly. Get ready to master this essential skill and unlock your potential for efficient, organized thinking.The-Science-of-Goal-Setting-and-Achieving-Success
The-Science-of-Goal-Setting-and-Achieving-SuccessOzias Rondon
油
Unlock the power of goal setting with science-backed strategies to turn your dreams into reality. Learn how to set SMART goals, stay motivated, and overcome obstacles to achieve success in any area of life.
The paradox of Perfection - why performance pressure counteracts leadership
The paradox of Perfection - why performance pressure counteracts leadershipWico Mulder
油
Global Leadership Platform - April 5th 2025 - Kasteel de Vanenburg (Oragnization Ellen Tuenter)Story, Life Essences Intervention Program
Story, Life Essences Intervention ProgramOmar Al Naqa
油
Dedicated to The First Flower After Winter
I love you without knowing how, or when, or from where. I love you simply, without problems or pride: I love you in this way because I do not know any other way of loving but this, in which there is no I or you, so intimate that your hand upon my chest is my hand, so intimate that when I fall asleep your eyes close. Pablo Neruda
STORY starts with gifts from the sky into your ability to love and how you choose to treat people.
"Your beliefs dont make you a better person, your behavior does." You are You storyHow Yoga Nidra Teacher Training Prepares You for Spiritual Growth
How Yoga Nidra Teacher Training Prepares You for Spiritual GrowthDavid Cook
油
Yoga Nidra, often referred to as yogic sleep, is a transformative practice of deep relaxation and inner realization. Incorporating yoga nidra practices into your daily routine helps strengthen both personal and spiritual growth, creating a deeper connection to your inner self.
The dev project
- 1. By Matt
- 3. To solve this problem you have to get the numerator and denominator to simplest forms Since there are four proportional terms we can group the numerator so we start of by factoring out the GCF After doing that we end up with this because both sides are multiplied by (x+9) Because of difference of squares we are able to simplify it even more
- 4. Since the denominator is not proportional we must long divide using -6(given) To long divide you must find out what you have to multiply by x to get x続( its x族) then multiply that by the rest of the polynomial then subtract the new from the old and repeat We come out with this Reduced to
- 5. Now we must get what we need for the graph out of the equations Factored Standard Vertical Asymptote: -6,7 in denominator of factor form when we put 0 in for x Horizontal Asymptote: 1 in standard form taking highest power then dividing X int: -9,8 in numerator in factor form when we put 0 in for x Y int: 12/7 taking to lowest powers from standard form and dividing Hole: -8 in factored form when numerator and denominator are the same
- 6. The Graph
- 7. Question 2 Mark has 200 feet of fence he want to make a fence around his garden and use his house as one of the sides to make the garden bigger. What is the maximum area Mark can have? Find domain and range when done
- 8. We must start of by making 2 equations that show perimeter and area Since we have 200 feet of fence and we have 3 sides 2 of which are the same the equation is 200=2x+y
- 9. We must solve for a now Start of by solving the Perimeter formula 200-2x=y Put this equation in for y in the area equation A=x(200-2x) A= -2x族+200x
- 10. To find the maximum value use the formula -b/2a to find the x of the vertex -200/-4= 50 Plug x into the equation to get the maximum -2(50)族+200(50)=5000 D:(-,) No number x cannot equal R:(-,5000] No minimum value and already found the maximum
- 12. Start of by distributing (x+4) to (x-4) Then you should get Which simplifies to
- 13. Distribute (x族-16) to (x+6) You should get
- 14. Distribute (x続+6x族-16x-96) to (x-10) You should come out with Simplify to
- 15. Distribute (x-4x続-76x族-256x+960) to (x+2) Comes out to Simplifies to
- 17. Get it Equal to Zero Subtract x族 then add -2x Which should get you 9x族+84x+27=0
- 18. Set it up (9x+ )( + ) You must then find to numbers that multiply to get 27 but add to get 84x
- 19. You should come up with (9x+3)(x+9) X=3/9 X=9