To solve the algebra equation x+1=3 using alge-tiles, subtract 1 from both sides of the equation to get x+1-1=3-1, then cancel out the 1 and -1 to make zero pairs and remove them, leaving x=2 as the solution.
1. To solve equations with radicals, isolate the radical term by adding or subtracting like terms, then square both sides of the equation.
2. The "u" substitution method can be used to solve quadratic equations in radical form by letting the variable under the radical equal "u" and squaring it.
3. After substitution and factoring, the solutions for "u" are then substituted back in terms of the original variable under the radical.
This document provides an overview of algebraic expressions and polynomials from a Math 1000 course. It introduces polynomials as sums or differences of monomials, which are variables raised to exponents. Examples of adding, subtracting, and multiplying polynomials are shown. The key processes of factoring trinomials and special factoring formulas are also explained in detail. Factoring is emphasized as an important math skill.
This document provides steps for solving rational equations:
1) Find the least common denominator (LCD) of all terms in the equation.
2) Multiply both sides of the equation by the LCD.
3) Solve the resulting equation.
4) Check that any solutions satisfy the original equation, as some solutions may be "extraneous roots" that make the denominator equal to zero.
The document includes examples demonstrating these steps, such as solving equations with factored denominators and equations where cross-multiplying eliminates the fractions.
1. The document discusses algebraic expressions and operations such as addition, subtraction, multiplication, and division of algebraic expressions. It provides examples and step-by-step solutions to problems involving these operations.
2. Key concepts covered include the properties and rules of exponents, signs, and how to perform operations on monomials, polynomials, and algebraic expressions. Special products like the difference and sum of cubes are also explained.
3. Worked examples provide solutions to problems involving adding, subtracting, multiplying, and dividing algebraic monomials and polynomials. Concepts like factoring trinomials using special products are demonstrated through examples.
I have added to the original presentation in response to one of the comments.... the result of 'x' is correct on slide 7, take a look at the new version of this ppt to clear up any confusion about why...
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 different types of equations and methods for solving them. It begins by defining what an equation is - a mathematical statement with two expressions separated by an equal sign.
It then describes linear equations as having degree one, quadratic equations as having degree two, and so on. The document provides examples of each type of equation.
Finally, it outlines the steps for solving different types of equations. It discusses removing parentheses and denominators, grouping terms, and using formulas to solve linear, incomplete quadratic, and quadratic equations. Examples are provided to illustrate each solving method.
The document discusses various algebraic expressions and operations including:
1) Finding the numeric value of algebraic expressions by substituting values for variables and simplifying.
2) Adding, subtracting, multiplying, and dividing algebraic expressions using properties like the distributive property.
3) Factoring expressions using factoring by grouping, difference of squares, and perfect square trinomials.
1. To solve equations with radicals, isolate the radical term by adding or subtracting like terms, then square both sides of the equation.
2. The "u" substitution method can be used to solve quadratic equations in radical form by letting the variable under the radical equal "u" and squaring it.
3. After substitution and factoring, the solutions for "u" are then substituted back in terms of the original variable under the radical.
This document provides an overview of algebraic expressions and polynomials from a Math 1000 course. It introduces polynomials as sums or differences of monomials, which are variables raised to exponents. Examples of adding, subtracting, and multiplying polynomials are shown. The key processes of factoring trinomials and special factoring formulas are also explained in detail. Factoring is emphasized as an important math skill.
This document provides steps for solving rational equations:
1) Find the least common denominator (LCD) of all terms in the equation.
2) Multiply both sides of the equation by the LCD.
3) Solve the resulting equation.
4) Check that any solutions satisfy the original equation, as some solutions may be "extraneous roots" that make the denominator equal to zero.
The document includes examples demonstrating these steps, such as solving equations with factored denominators and equations where cross-multiplying eliminates the fractions.
1. The document discusses algebraic expressions and operations such as addition, subtraction, multiplication, and division of algebraic expressions. It provides examples and step-by-step solutions to problems involving these operations.
2. Key concepts covered include the properties and rules of exponents, signs, and how to perform operations on monomials, polynomials, and algebraic expressions. Special products like the difference and sum of cubes are also explained.
3. Worked examples provide solutions to problems involving adding, subtracting, multiplying, and dividing algebraic monomials and polynomials. Concepts like factoring trinomials using special products are demonstrated through examples.
I have added to the original presentation in response to one of the comments.... the result of 'x' is correct on slide 7, take a look at the new version of this ppt to clear up any confusion about why...
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 different types of equations and methods for solving them. It begins by defining what an equation is - a mathematical statement with two expressions separated by an equal sign.
It then describes linear equations as having degree one, quadratic equations as having degree two, and so on. The document provides examples of each type of equation.
Finally, it outlines the steps for solving different types of equations. It discusses removing parentheses and denominators, grouping terms, and using formulas to solve linear, incomplete quadratic, and quadratic equations. Examples are provided to illustrate each solving method.
The document discusses various algebraic expressions and operations including:
1) Finding the numeric value of algebraic expressions by substituting values for variables and simplifying.
2) Adding, subtracting, multiplying, and dividing algebraic expressions using properties like the distributive property.
3) Factoring expressions using factoring by grouping, difference of squares, and perfect square trinomials.
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 ± b)^3 and taking the square of trinomial
1 2 2nd-degree equation and word problems-xmath123b
This document discusses solving polynomial equations by factoring. It provides an example of solving the equation x^2 - 2x - 3 = 0 by: 1) factoring the trinomial to get (x - 3)(x + 1) = 0, 2) setting each factor equal to 0 to get x - 3 = 0 and x + 1 = 0, and 3) extracting the solutions x = 3 and x = -1. It then works through another example of solving 2 = 2x^2 - 3x by similar factoring steps to get the solutions x = -1/2, x = 2. Finally, it introduces word problems that can be modeled by quadratic equations of the form AB = C.
The document contains information about partial fraction decomposition:
1. It discusses four cases for partial fraction decomposition based on the factors of the denominator: distinct linear factors, repeated linear factors, distinct irreducible quadratic factors, and repeated irreducible quadratic factors.
2. It provides examples to illustrate each case, showing how to set up and solve systems of equations to determine the coefficients of the partial fractions.
3. Homework Task 4 on systems of equations and inequalities is due on August 13, and consultation times for Ms. Durandt are on Thursdays and Fridays at 10:30.
1) The limit of (1 + 1/n)^n as n approaches infinity is e, which is approximately equal to 2.71828.
2) Taking the binomial expansion of (1 + x/n)^n and taking the limit as n approaches infinity yields the power series for ex.
3) The natural logarithm ln(x) is defined as the inverse function of ex. Its integral form is derived from the differential equation d(ln(x))/dx = 1/x.
A quadratic equation is an equation equivalent to the form ax^2 + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. To solve a quadratic equation, we first get it into standard form and then either factor if possible or use the quadratic formula. If factoring results in a negative number under a square root, there are no real solutions. Completing the square is another method that allows us to factor a quadratic expression and solve for the roots.
A quadratic equation is a polynomial equation of the second degree. It can be written in the general form ax^2 + bx + c = 0, where a, b, and c are coefficients. The quadratic formula x = (-b ± √(b^2 - 4ac))/2a can be used to find the roots (solutions) of a quadratic equation. The discriminant, Δ = b^2 - 4ac, determines whether the roots are real numbers, a repeated real root, or complex numbers. Descartes' rule of signs relates the number of variations and permanences in the signs of the coefficients a, b, and c to the number of positive and negative roots.
The document discusses algebraic expressions and provides examples of operations on different types of expressions. It begins by defining algebraic expressions and their components such as terms, monomials, binomials, trinomials, and polynomials. It then provides examples of addition, subtraction, multiplication, and division of monomials, binomials, trinomials, and polynomials. Specific steps are outlined for performing each operation to obtain the correct algebraic expression as the result. Examples of evaluating algebraic expressions for given numeric values of variables are also provided.
Steps are shown for solving systems of equations by the elimination method, including multiplying one equation by a constant to eliminate a variable before adding the equations. Examples are provided of solving systems by substitution, where one variable is solved for in one equation and substituted into the other equation. Practice problems are given for students to apply
The document discusses parabolas and their real world applications. It provides the vertex and equation of a parabolic curve with a vertex of (3,3). The equation y=a(x-h)^2+k is used to calculate the final equation of y=-1/8(x-3)^2+3 for the parabola. The key properties are that the parabola opens down and the equation contains negative numbers with values close to the calculated equation of y= -0.125x^2+0.75x+1.125+3.
This document discusses solving two-step equations and provides examples of solving equations such as 2x + 1 = 4, X - 2 = 3 + 3, X + 2x = 6, and -x + 4 = 10. The solutions to these example equations are provided as x = 3/2, x = 8, x = 2, and x = -6, respectively.
Rational expressions are fractions where the numerator and denominator are polynomials. To simplify rational expressions, we first factor the polynomials and then cancel any common factors. Adding and subtracting rational expressions follows the same process as fractions - find the least common denominator, multiply the numerators and denominators to get the same denominator, then add or subtract the numerators. Multiplying rational expressions involves factoring and cancelling common factors between the numerator and denominator. To divide rational expressions, we multiply the first expression by the reciprocal of the second expression and then factor and cancel. Word problems involving rational expressions can be solved by identifying what is known and unknown, setting up equations relating the known and unknown values, and then solving the equations.
This document provides examples of solving linear equations by manipulating the equations to isolate the variable. It shows adding or subtracting the same quantity from both sides, distributing terms, combining like terms, and dividing both sides by a coefficient to isolate the variable. The examples are worked through step-by-step and checked by substitution back into the original equation.
The document provides examples and explanations for solving simple linear equations and inequalities over rational numbers. It includes examples of determining if a number is a solution to an equation by substitution, using inverse operations like addition and subtraction to isolate variables, and applying equations to word problems involving forces. Step-by-step workings are shown for solving equations and checking solutions.
1) A linear equation in two variables is an equation of the form ax + by + c = 0, where a, b, and c are real numbers and a and b are not both zero. The graph of a linear equation is a straight line.
2) There are two main methods to solve a pair of linear equations: the substitution method and the elimination method. The substitution method involves solving one equation for one variable in terms of the other and substituting it into the other equation. The elimination method involves multiplying the equations by constants to eliminate one variable.
3) Solving a pair of linear equations using these methods will result in either a unique solution, infinitely many solutions, or no solution, depending
The document discusses algebraic fractions and rational expressions. It provides instructions for how to:
1) Simplify and reduce rational expressions by factoring and dividing common factors in the numerator and denominator.
2) Multiply and divide rational expressions by multiplying or dividing the numerators and denominators.
3) Add and subtract rational expressions by finding a common denominator and combining like terms in the numerator.
4) Solve rational equations by clearing fractions, multiplying both sides by the least common denominator, and isolating the variable.
This document discusses using recurrence relations to model problems involving counting techniques. It provides examples of modeling problems related to bacteria population growth, rabbit population growth, the Tower of Hanoi puzzle, and valid codeword enumeration. For each problem, it defines the recurrence relation and initial conditions, derives a closed-form solution, and proves its correctness using mathematical induction. Recurrence relations provide a way to define sequences and solve problems recursively by relating terms to previous terms in the sequence.
O documento é uma apostila sobre contabilidade geral para concursos públicos que apresenta conceitos básicos da contabilidade, como definição, objetivos, técnicas e princípios. Aborda também a função administrativa e econômica da contabilidade no controle patrimonial e apuração de resultados.
The document discusses color and font choices for a school publication. It tests different background colors like blue, red, pink, and brown before settling on a neutral paper grain texture. Yellow and pink outlines are tried before matching the outlines to the brown background. Different fonts are sampled for the masthead before choosing a bold, mature style. Header colors like yellow and gray are deemed too bright or boring, with light aqua chosen as an eye-catching option that blends well. The font Corbel is selected for body text as a readable option that will appeal to varied audiences.
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 ± b)^3 and taking the square of trinomial
1 2 2nd-degree equation and word problems-xmath123b
This document discusses solving polynomial equations by factoring. It provides an example of solving the equation x^2 - 2x - 3 = 0 by: 1) factoring the trinomial to get (x - 3)(x + 1) = 0, 2) setting each factor equal to 0 to get x - 3 = 0 and x + 1 = 0, and 3) extracting the solutions x = 3 and x = -1. It then works through another example of solving 2 = 2x^2 - 3x by similar factoring steps to get the solutions x = -1/2, x = 2. Finally, it introduces word problems that can be modeled by quadratic equations of the form AB = C.
The document contains information about partial fraction decomposition:
1. It discusses four cases for partial fraction decomposition based on the factors of the denominator: distinct linear factors, repeated linear factors, distinct irreducible quadratic factors, and repeated irreducible quadratic factors.
2. It provides examples to illustrate each case, showing how to set up and solve systems of equations to determine the coefficients of the partial fractions.
3. Homework Task 4 on systems of equations and inequalities is due on August 13, and consultation times for Ms. Durandt are on Thursdays and Fridays at 10:30.
1) The limit of (1 + 1/n)^n as n approaches infinity is e, which is approximately equal to 2.71828.
2) Taking the binomial expansion of (1 + x/n)^n and taking the limit as n approaches infinity yields the power series for ex.
3) The natural logarithm ln(x) is defined as the inverse function of ex. Its integral form is derived from the differential equation d(ln(x))/dx = 1/x.
A quadratic equation is an equation equivalent to the form ax^2 + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. To solve a quadratic equation, we first get it into standard form and then either factor if possible or use the quadratic formula. If factoring results in a negative number under a square root, there are no real solutions. Completing the square is another method that allows us to factor a quadratic expression and solve for the roots.
A quadratic equation is a polynomial equation of the second degree. It can be written in the general form ax^2 + bx + c = 0, where a, b, and c are coefficients. The quadratic formula x = (-b ± √(b^2 - 4ac))/2a can be used to find the roots (solutions) of a quadratic equation. The discriminant, Δ = b^2 - 4ac, determines whether the roots are real numbers, a repeated real root, or complex numbers. Descartes' rule of signs relates the number of variations and permanences in the signs of the coefficients a, b, and c to the number of positive and negative roots.
The document discusses algebraic expressions and provides examples of operations on different types of expressions. It begins by defining algebraic expressions and their components such as terms, monomials, binomials, trinomials, and polynomials. It then provides examples of addition, subtraction, multiplication, and division of monomials, binomials, trinomials, and polynomials. Specific steps are outlined for performing each operation to obtain the correct algebraic expression as the result. Examples of evaluating algebraic expressions for given numeric values of variables are also provided.
Steps are shown for solving systems of equations by the elimination method, including multiplying one equation by a constant to eliminate a variable before adding the equations. Examples are provided of solving systems by substitution, where one variable is solved for in one equation and substituted into the other equation. Practice problems are given for students to apply
The document discusses parabolas and their real world applications. It provides the vertex and equation of a parabolic curve with a vertex of (3,3). The equation y=a(x-h)^2+k is used to calculate the final equation of y=-1/8(x-3)^2+3 for the parabola. The key properties are that the parabola opens down and the equation contains negative numbers with values close to the calculated equation of y= -0.125x^2+0.75x+1.125+3.
This document discusses solving two-step equations and provides examples of solving equations such as 2x + 1 = 4, X - 2 = 3 + 3, X + 2x = 6, and -x + 4 = 10. The solutions to these example equations are provided as x = 3/2, x = 8, x = 2, and x = -6, respectively.
Rational expressions are fractions where the numerator and denominator are polynomials. To simplify rational expressions, we first factor the polynomials and then cancel any common factors. Adding and subtracting rational expressions follows the same process as fractions - find the least common denominator, multiply the numerators and denominators to get the same denominator, then add or subtract the numerators. Multiplying rational expressions involves factoring and cancelling common factors between the numerator and denominator. To divide rational expressions, we multiply the first expression by the reciprocal of the second expression and then factor and cancel. Word problems involving rational expressions can be solved by identifying what is known and unknown, setting up equations relating the known and unknown values, and then solving the equations.
This document provides examples of solving linear equations by manipulating the equations to isolate the variable. It shows adding or subtracting the same quantity from both sides, distributing terms, combining like terms, and dividing both sides by a coefficient to isolate the variable. The examples are worked through step-by-step and checked by substitution back into the original equation.
The document provides examples and explanations for solving simple linear equations and inequalities over rational numbers. It includes examples of determining if a number is a solution to an equation by substitution, using inverse operations like addition and subtraction to isolate variables, and applying equations to word problems involving forces. Step-by-step workings are shown for solving equations and checking solutions.
1) A linear equation in two variables is an equation of the form ax + by + c = 0, where a, b, and c are real numbers and a and b are not both zero. The graph of a linear equation is a straight line.
2) There are two main methods to solve a pair of linear equations: the substitution method and the elimination method. The substitution method involves solving one equation for one variable in terms of the other and substituting it into the other equation. The elimination method involves multiplying the equations by constants to eliminate one variable.
3) Solving a pair of linear equations using these methods will result in either a unique solution, infinitely many solutions, or no solution, depending
The document discusses algebraic fractions and rational expressions. It provides instructions for how to:
1) Simplify and reduce rational expressions by factoring and dividing common factors in the numerator and denominator.
2) Multiply and divide rational expressions by multiplying or dividing the numerators and denominators.
3) Add and subtract rational expressions by finding a common denominator and combining like terms in the numerator.
4) Solve rational equations by clearing fractions, multiplying both sides by the least common denominator, and isolating the variable.
This document discusses using recurrence relations to model problems involving counting techniques. It provides examples of modeling problems related to bacteria population growth, rabbit population growth, the Tower of Hanoi puzzle, and valid codeword enumeration. For each problem, it defines the recurrence relation and initial conditions, derives a closed-form solution, and proves its correctness using mathematical induction. Recurrence relations provide a way to define sequences and solve problems recursively by relating terms to previous terms in the sequence.
O documento é uma apostila sobre contabilidade geral para concursos públicos que apresenta conceitos básicos da contabilidade, como definição, objetivos, técnicas e princípios. Aborda também a função administrativa e econômica da contabilidade no controle patrimonial e apuração de resultados.
The document discusses color and font choices for a school publication. It tests different background colors like blue, red, pink, and brown before settling on a neutral paper grain texture. Yellow and pink outlines are tried before matching the outlines to the brown background. Different fonts are sampled for the masthead before choosing a bold, mature style. Header colors like yellow and gray are deemed too bright or boring, with light aqua chosen as an eye-catching option that blends well. The font Corbel is selected for body text as a readable option that will appeal to varied audiences.
1) Los estudiantes están más satisfechos con su decisión de estudiar pedagogía en inglés que con la infraestructura y recursos disponibles.
2) Consideran que las relaciones con los profesores son buenas, pero podrían mejorar la relación con el jefe de carrera.
3) Aunque los estudiantes están comprometidos con sus estudios, la infraestructura deficiente y el momento tardío de la práctica profesional los afectan negativamente.
C:\Fakepath\Production Stages Of 2nd Half Of Double Pageguestb68d13
The document outlines the production stages for the second half of a double page spread which includes adding two images, questions and answers from an interview, and a closing summative comment alongside a floating quote.
Wilson Hernando Cely Jimenez outlines his life goals which include becoming a professional mechanic in one year by working in the Colombian Air Force, becoming a professional DJ in five years, getting married and having two children with his wife in seven years, going on vacation to Argentina, buying a new car, and purchasing a new house in Ibagué.
Este documento explora la relación entre las prácticas sociales del lenguaje y los medios de comunicación. Discute cómo los medios influyen en el lenguaje al enriquecer el vocabulario y cómo las personas tienden a comunicarse de acuerdo a lo que ven y escuchan. También analiza cómo diferentes contextos y culturas modifican el lenguaje y cómo los locutores deben considerar la variación lingüística de su audiencia. Concluye que el lenguaje y los medios están estrechamente relacionados y que es importante seleccionar
This document contains a list of four names of students who are members of a group: Maria milagro camacaro, Angelica Colmenárez, Skarlet Delgado, and Liz Peña. It does not provide any other details about the purpose of the group or its activities.
This document contains the contact information and publications of Eric J. Weigel. It lists his address, email, phone number and then provides a list of journal articles and recent articles he has authored or co-authored on topics related to foreign exchange returns, bond/call option strategies, stock market volatility, inflation dynamics, market timing skills, index membership effects, global equity and fixed income strategies, tactical asset allocation, relationships between stock returns in different markets, and financial instruments. The recent articles provide a sampling of his work on topics like volatility, non-European stock exposure, searching for yield, combining alternative and traditional strategies, disentangling industry and country effects, growth and value returns, the role of quality, commodities
The document lists a variety of activities that people can do such as write on a board, type on a computer, play skateboard, read a book, go on a picnic, play golf, grill food, go to school, and go upstairs.
O documento apresenta sete princípios para manter o cérebro saudável e produtivo: 1) fazer exercícios físicos diariamente; 2) beber pelo menos oito copos de água por dia; 3) realizar exercícios de respiração e se expor à luz do sol e plantas; 4) consumir alimentos benéficos ao cérebro como frutas, verduras e peixe; 5) manter pensamentos positivos; 6) escutar música; 7) estimular o cérebro com atividades mentais como jogos
Los conjuntos A, B y C se utilizan para ilustrar dos problemas de operaciones con conjuntos, incluyendo la intersección de B y C y la unión de A con la intersección de A y C.
Rubras mariposas revisita o apogeu e decadência de uma outra londrina londr...nadiaval
O curta-metragem Rubras Mariposas, dirigido por Anderson Craveiro, retrata a Londrina dos anos 1950 e 1970, desde o auge da produção cafeeira até seu declínio. O filme conta com mais de 100 figurantes e atores renomados como Mário Bortoloto, Maria Alice Vergueiro e Luciano Chirolli. A inspiração veio do livro "Terra Vermelha" e a produção deve estrear em dezembro.
Este álbum de fotografías documenta un viaje por Madrid, la capital de España. Captura imágenes de lugares emblemáticos como la Plaza Mayor y el Palacio Real, así como escenas cotidianas de la vida en la ciudad. El álbum ofrece una mirada visual a la cultura y la belleza de Madrid.
The document appears to be a collection of random letters and symbols with no clear meaning or message. It does not contain enough coherent information to generate a multi-sentence summary.
House of Pain fue un grupo de hip hop estadounidense de los años 90 liderado por Everlast. Sacaron 3 álbumes antes de disolverse, siendo especialmente conocidos por el éxito "Jump Around". Everlast inició una carrera en solitario tras la ruptura mientras los otros miembros también continuaron trabajando en el mundo de la música.