Factoring Pt. 2/2 (x^2 + bx + c)
•Download as PPTX, PDF•
0 likes•374 views
1. The document discusses factoring quadratic expressions using the greatest common factor (GCF) method. 2. It provides examples of factoring expressions such as 3x^2 + 8x + 4 by trying different pairs of binomials that multiply to the GCF and constant term. 3. The document also addresses factoring expressions with negative coefficients, explaining that you can factor the expression inside the parentheses and multiply the whole expression by -1.
Report
Share
Report
Share
Recommended
Factoring pt. 1/2
1. The document provides step-by-step instructions for factoring quadratic expressions using reverse FOIL. It explains that the factors are determined by the constant term's factors and shows examples of factoring expressions with positive and negative coefficients.
2. Problems include factoring expressions into standard form, factoring expressions with variables, and solving a word problem involving the area of two shapes to determine the width of one. The document breaks problems down into clear steps and provides answers on subsequent slides.
Multiplying & dividing rational expressions
This document provides instruction on multiplying, dividing, and simplifying rational expressions. It begins with mini lessons on multiplying fractions and simplifying fractions before and after multiplying. It then works through 4 example problems, showing the step-by-step work of multiplying and simplifying rational expressions. The last section provides a mini lesson on dividing fractions and works through an example of dividing two rational expressions.
Adding & subtracting rational expressions
This document provides an overview of adding and subtracting rational expressions. It begins with instructions on finding the least common denominator (LCD) and provides examples of adding fractions with unlike denominators. It then demonstrates subtracting fractions with unlike denominators by first making the denominators the same. Several examples of adding and subtracting rational expressions are worked through step-by-step. Special cases involving factoring denominators are also discussed.
Solving Linear Equations
This document summarizes three methods for solving systems of linear equations: graphing, substitution, and elimination. It provides examples of solving systems of two equations using each method. Graphing involves plotting the lines defined by each equation on a coordinate plane and finding their point of intersection. Substitution involves isolating a variable in one equation and substituting it into the other equation. Elimination involves adding or subtracting multiples of equations to remove a variable and solve for the remaining variable.
Solve for x in terms of y
(1) The document provides instructions to solve equations for x in terms of y. (2) As an example, it shows solving the equation 3xy = 8 for x, which results in the expression x = 8/(3y). (3) It also shows solving the equation 1+y/2+x = 3y for x, which gives the solution x = (1-5y)/(3y).
Tut 1
This document contains 11 questions regarding differential equations. It asks the student to classify equations as order, linear or non-linear; formulate differential equations by eliminating constants; solve specific differential equations; prove properties of curves; find general solutions by substitution; and determine conditions for exactness and solve exact differential equations.
Algebra Lineal. Tarea1
This document provides 10 systems of linear equations and asks to find the solution to each system. It lists the equations for 10 different systems of linear equations, with the unknowns x1, x2 and x3. The task is to solve for the values of the unknowns that satisfy all equations simultaneously in each system.
Order of operations basic practice
The document provides instructions on using the order of operations PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction) to evaluate numerical expressions. It gives examples of solving expressions step-by-step using PEMDAS. Students are then given 10 practice problems and instructed to use PEMDAS and a calculator, if available, to find the value of each expression.
Recommended
Factoring pt. 1/2
1. The document provides step-by-step instructions for factoring quadratic expressions using reverse FOIL. It explains that the factors are determined by the constant term's factors and shows examples of factoring expressions with positive and negative coefficients.
2. Problems include factoring expressions into standard form, factoring expressions with variables, and solving a word problem involving the area of two shapes to determine the width of one. The document breaks problems down into clear steps and provides answers on subsequent slides.
Multiplying & dividing rational expressions
This document provides instruction on multiplying, dividing, and simplifying rational expressions. It begins with mini lessons on multiplying fractions and simplifying fractions before and after multiplying. It then works through 4 example problems, showing the step-by-step work of multiplying and simplifying rational expressions. The last section provides a mini lesson on dividing fractions and works through an example of dividing two rational expressions.
Adding & subtracting rational expressions
This document provides an overview of adding and subtracting rational expressions. It begins with instructions on finding the least common denominator (LCD) and provides examples of adding fractions with unlike denominators. It then demonstrates subtracting fractions with unlike denominators by first making the denominators the same. Several examples of adding and subtracting rational expressions are worked through step-by-step. Special cases involving factoring denominators are also discussed.
Solving Linear Equations
This document summarizes three methods for solving systems of linear equations: graphing, substitution, and elimination. It provides examples of solving systems of two equations using each method. Graphing involves plotting the lines defined by each equation on a coordinate plane and finding their point of intersection. Substitution involves isolating a variable in one equation and substituting it into the other equation. Elimination involves adding or subtracting multiples of equations to remove a variable and solve for the remaining variable.
Solve for x in terms of y
(1) The document provides instructions to solve equations for x in terms of y. (2) As an example, it shows solving the equation 3xy = 8 for x, which results in the expression x = 8/(3y). (3) It also shows solving the equation 1+y/2+x = 3y for x, which gives the solution x = (1-5y)/(3y).
Tut 1
This document contains 11 questions regarding differential equations. It asks the student to classify equations as order, linear or non-linear; formulate differential equations by eliminating constants; solve specific differential equations; prove properties of curves; find general solutions by substitution; and determine conditions for exactness and solve exact differential equations.
Algebra Lineal. Tarea1
This document provides 10 systems of linear equations and asks to find the solution to each system. It lists the equations for 10 different systems of linear equations, with the unknowns x1, x2 and x3. The task is to solve for the values of the unknowns that satisfy all equations simultaneously in each system.
Order of operations basic practice
The document provides instructions on using the order of operations PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction) to evaluate numerical expressions. It gives examples of solving expressions step-by-step using PEMDAS. Students are then given 10 practice problems and instructed to use PEMDAS and a calculator, if available, to find the value of each expression.
Section2 stochastic
1. The document discusses concepts related to expectation and variance of random variables including expected value, variance, moments, and examples of calculating these for different probability distributions like uniform, normal, exponential, and Rayleigh.
2. Problems at the end provide examples of computing expected value, variance, and cumulative distribution function for random variables following different distributions. Solutions show the calculations and formulas used.
3. Key formulas introduced include definitions of expected value and variance, the relationship between them, and formulas for calculating moments, expected value, and variance for specific distributions. Examples demonstrate applying the concepts and formulas to problems.
Integral tentu fungsi aljabar
This document is an 8-page mathematics module on integral calculus in Bahasa Indonesia. It provides instructions on how to study the module, lists the learning objectives and indicators, and gives examples of determining the definite integrals of algebraic functions. It explains the basic concepts of definite integrals, such as their formula and properties. Several worked examples are shown to illustrate how to calculate the definite integral of various algebraic expressions between given bounds. Students are assigned practice problems to solve.
AA Section 6-5
1. The document provides steps for completing the square of a quadratic equation to convert it from standard form to vertex form. It shows how to isolate the x terms, find the value of b, add (1/2b)2 to both sides, and factor to obtain the vertex form.
2. An example problem walks through rewriting y = x2 + 18x + 90 in vertex form by following the steps: isolating x terms, finding b, adding (1/2b)2, and factoring the perfect square trinomial to obtain the vertex form y - 9 = (x + 9)2.
3. Completing the square is a method to convert a quadratic equation from standard form
C6 6.5
The document contains solutions to 26 math problems involving multiplying polynomials. The problems involve multiplying terms with variables like x, y, a, b. The solutions show the distribution of terms and combining like terms. Some problems then verify the solutions by plugging in values for the variables. The document provides worked out solutions to multiplying polynomials of varying complexities.
3m. Pedagogy of Mathematics (Part II) - Algebra (Ex 3.13)
Pedagogy of Mathematics (Part II) - Algebra, Algebra, Maths, IX std Maths, Samacheerkalvi maths, II year B.Ed., Pedagogy, Mathematics, solving by cross multiplication method
Engineering mathematics divergence
The document presents a summary of a presentation on engineering mathematics. It defines the divergence of a vector function as the sum of the partial derivatives of the components of the vector function with respect to x, y and z. As an example, it calculates the divergence of the vector function φ=ix^2+jy^2+kz^2 at the point (1,0,1), obtaining a result of 4.
Engr 213 midterm 2a 2009
This document is an exam for a differential equations course. It contains 4 problems to solve in 60 minutes. Problem 1 asks students to determine if functions are linearly dependent or independent on an interval. Problem 2 is an initial value problem to solve. Problem 3 is to solve a differential equation. Problem 4 is to find the general solution of another differential equation. Formulas for trigonometric integrals are provided.
第二次作業
1. The document proves through mathematical induction that the formula nj=1 j^2 = n(n+1)(2n+1)/6 for the sum of squares from 1 to n is true for all positive integers n.
2. It also proves that a 2n x 2n chessboard with one square missing can be covered with L-shaped pieces that each cover three squares, again using mathematical induction.
3. Mathematical induction is used to prove both formulas by showing the base case holds and assuming the formula is true for an integer n to prove it is true for n+1.
Fracciones algeb producto_cociente_blog
This document provides examples of operations with algebraic fractions, including addition, subtraction, multiplication, and division. It shows the steps to perform these operations and simplify the results. Some examples include combining like terms and factorizing expressions. The goal is to teach learners how to manipulate algebraic fractions algebraically.
Simplifying Basic Rational Expressions Part 1
This document provides instructions and examples for simplifying rational expressions by factoring and canceling terms. It begins with definitions of rational expressions and undefined values. Examples shown include finding excluded values that make expressions undefined, as well as simplifying expressions by factoring numerators and denominators and canceling common factors. The goal is to either find excluded values or simplify the expressions into their lowest terms. Ten problems with step-by-step solutions are provided to demonstrate the techniques.
Multiplying Binomials and Trinomials
This document provides steps for multiplying a binomial and a trinomial:
1) Make a table with the terms of each expression.
2) Multiply corresponding terms of the table.
3) Write the products in order from highest to lowest exponent.
4) Simplify by combining like terms.
Ecuaciones
This document contains 32 systems of equations and their solutions. The systems include linear equations, quadratic equations, and equations containing variables multiplied together. Solving the systems requires skills like adding or subtracting equations, substituting values, and solving quadratics. The solutions are provided in fractional or decimal form depending on the system.
Week 3 [compatibility mode]
The document discusses methods for solving first order ordinary differential equations (ODEs). It covers:
1) Finding the integrating factor for exact differential equations.
2) Solving homogeneous first order linear ODEs by making a substitution to reduce it to a separable equation.
3) Solving inhomogeneous first order linear ODEs using an integrating factor.
Examples are provided to demonstrate each method step-by-step.
0309 ch 3 day 9
This document discusses rational functions and their asymptotes. It begins by stating to predict all asymptotes and graph rational functions to verify the asymptotes. It then provides examples of rational functions and shows how to find their vertical, horizontal and slant asymptotes. It demonstrates dividing the polynomials of a rational function to find the slant asymptote. It concludes by analyzing the end behavior of rational functions and stating that the slant asymptote is found using the quotient polynomial.
Activity math9
The document contains solutions to two quadratic equations. The first equation is solved by factorizing into (x - 1)2 = 4, giving solutions of x = 3 and x = -1. The second equation is completed by completing the square, giving (x + 5/2)2 = 121/4 and solutions of x = 3 and x = -8.
Systems Non Linear Equations
The document discusses systems of non-linear equations and their properties. It covers different types of non-linear equations like absolute value, quadratic, and their graph shapes. It provides tips for graphing systems of non-linear equations on a calculator, such as making sure the view captures all intersections and using trace to locate the intersections when there are multiple. Examples of specific non-linear equation systems are also given.
Sect1 R
This document provides 36 review problems to identify the type of first-order differential equations and appropriate solution methods discussed in Chapter 1. The problems cover linear equations, separable equations, homogeneous equations, exact equations, and Bernoulli equations. An example solution is provided for several problems identifying the type of equation and corresponding general solution.
Chess board problem(divide and conquer)
The document describes the divide and conquer algorithm for solving the defective chessboard problem. The problem is to cover a chessboard of size n x n, where n is a power of 2, with L-shaped tiles (trominoes), except for one defective square. The algorithm divides the board recursively into equal half-sized subboards until reaching boards of size 2x2, which can be covered with at most one tile. It then combines the solutions to the subproblems to construct a solution for the original board. The time complexity of this divide and conquer algorithm is O(n^2).
Rational equations
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.
Factor by grouping
The document provides steps for factoring polynomial expressions:
1) Separate the expression into terms with common factors and group those terms;
2) Find the greatest common factor (GCF) of the grouped terms;
3) Enclose the GCF and the grouped terms in parentheses to arrive at the factored expression.
Worked examples demonstrate factoring various polynomials, including using negative signs properly.
Polynomial math
1) The document discusses various methods for manipulating and solving algebraic expressions, including adding, subtracting, and factoring polynomials.
2) Factoring techniques include grouping like terms, using the difference of squares formula, and recognizing perfect square trinomials.
3) The quadratic formula is introduced as a way to solve quadratic equations of the form ax2 + bx + c = 0.
More Related Content
What's hot
Section2 stochastic
1. The document discusses concepts related to expectation and variance of random variables including expected value, variance, moments, and examples of calculating these for different probability distributions like uniform, normal, exponential, and Rayleigh.
2. Problems at the end provide examples of computing expected value, variance, and cumulative distribution function for random variables following different distributions. Solutions show the calculations and formulas used.
3. Key formulas introduced include definitions of expected value and variance, the relationship between them, and formulas for calculating moments, expected value, and variance for specific distributions. Examples demonstrate applying the concepts and formulas to problems.
Integral tentu fungsi aljabar
This document is an 8-page mathematics module on integral calculus in Bahasa Indonesia. It provides instructions on how to study the module, lists the learning objectives and indicators, and gives examples of determining the definite integrals of algebraic functions. It explains the basic concepts of definite integrals, such as their formula and properties. Several worked examples are shown to illustrate how to calculate the definite integral of various algebraic expressions between given bounds. Students are assigned practice problems to solve.
AA Section 6-5
1. The document provides steps for completing the square of a quadratic equation to convert it from standard form to vertex form. It shows how to isolate the x terms, find the value of b, add (1/2b)2 to both sides, and factor to obtain the vertex form.
2. An example problem walks through rewriting y = x2 + 18x + 90 in vertex form by following the steps: isolating x terms, finding b, adding (1/2b)2, and factoring the perfect square trinomial to obtain the vertex form y - 9 = (x + 9)2.
3. Completing the square is a method to convert a quadratic equation from standard form
C6 6.5
The document contains solutions to 26 math problems involving multiplying polynomials. The problems involve multiplying terms with variables like x, y, a, b. The solutions show the distribution of terms and combining like terms. Some problems then verify the solutions by plugging in values for the variables. The document provides worked out solutions to multiplying polynomials of varying complexities.
3m. Pedagogy of Mathematics (Part II) - Algebra (Ex 3.13)
Pedagogy of Mathematics (Part II) - Algebra, Algebra, Maths, IX std Maths, Samacheerkalvi maths, II year B.Ed., Pedagogy, Mathematics, solving by cross multiplication method
Engineering mathematics divergence
The document presents a summary of a presentation on engineering mathematics. It defines the divergence of a vector function as the sum of the partial derivatives of the components of the vector function with respect to x, y and z. As an example, it calculates the divergence of the vector function φ=ix^2+jy^2+kz^2 at the point (1,0,1), obtaining a result of 4.
Engr 213 midterm 2a 2009
This document is an exam for a differential equations course. It contains 4 problems to solve in 60 minutes. Problem 1 asks students to determine if functions are linearly dependent or independent on an interval. Problem 2 is an initial value problem to solve. Problem 3 is to solve a differential equation. Problem 4 is to find the general solution of another differential equation. Formulas for trigonometric integrals are provided.
第二次作業
1. The document proves through mathematical induction that the formula nj=1 j^2 = n(n+1)(2n+1)/6 for the sum of squares from 1 to n is true for all positive integers n.
2. It also proves that a 2n x 2n chessboard with one square missing can be covered with L-shaped pieces that each cover three squares, again using mathematical induction.
3. Mathematical induction is used to prove both formulas by showing the base case holds and assuming the formula is true for an integer n to prove it is true for n+1.
Fracciones algeb producto_cociente_blog
This document provides examples of operations with algebraic fractions, including addition, subtraction, multiplication, and division. It shows the steps to perform these operations and simplify the results. Some examples include combining like terms and factorizing expressions. The goal is to teach learners how to manipulate algebraic fractions algebraically.
Simplifying Basic Rational Expressions Part 1
This document provides instructions and examples for simplifying rational expressions by factoring and canceling terms. It begins with definitions of rational expressions and undefined values. Examples shown include finding excluded values that make expressions undefined, as well as simplifying expressions by factoring numerators and denominators and canceling common factors. The goal is to either find excluded values or simplify the expressions into their lowest terms. Ten problems with step-by-step solutions are provided to demonstrate the techniques.
Multiplying Binomials and Trinomials
This document provides steps for multiplying a binomial and a trinomial:
1) Make a table with the terms of each expression.
2) Multiply corresponding terms of the table.
3) Write the products in order from highest to lowest exponent.
4) Simplify by combining like terms.
Ecuaciones
This document contains 32 systems of equations and their solutions. The systems include linear equations, quadratic equations, and equations containing variables multiplied together. Solving the systems requires skills like adding or subtracting equations, substituting values, and solving quadratics. The solutions are provided in fractional or decimal form depending on the system.
Week 3 [compatibility mode]
The document discusses methods for solving first order ordinary differential equations (ODEs). It covers:
1) Finding the integrating factor for exact differential equations.
2) Solving homogeneous first order linear ODEs by making a substitution to reduce it to a separable equation.
3) Solving inhomogeneous first order linear ODEs using an integrating factor.
Examples are provided to demonstrate each method step-by-step.
0309 ch 3 day 9
This document discusses rational functions and their asymptotes. It begins by stating to predict all asymptotes and graph rational functions to verify the asymptotes. It then provides examples of rational functions and shows how to find their vertical, horizontal and slant asymptotes. It demonstrates dividing the polynomials of a rational function to find the slant asymptote. It concludes by analyzing the end behavior of rational functions and stating that the slant asymptote is found using the quotient polynomial.
Activity math9
The document contains solutions to two quadratic equations. The first equation is solved by factorizing into (x - 1)2 = 4, giving solutions of x = 3 and x = -1. The second equation is completed by completing the square, giving (x + 5/2)2 = 121/4 and solutions of x = 3 and x = -8.
Systems Non Linear Equations
The document discusses systems of non-linear equations and their properties. It covers different types of non-linear equations like absolute value, quadratic, and their graph shapes. It provides tips for graphing systems of non-linear equations on a calculator, such as making sure the view captures all intersections and using trace to locate the intersections when there are multiple. Examples of specific non-linear equation systems are also given.
Sect1 R
This document provides 36 review problems to identify the type of first-order differential equations and appropriate solution methods discussed in Chapter 1. The problems cover linear equations, separable equations, homogeneous equations, exact equations, and Bernoulli equations. An example solution is provided for several problems identifying the type of equation and corresponding general solution.
Chess board problem(divide and conquer)
The document describes the divide and conquer algorithm for solving the defective chessboard problem. The problem is to cover a chessboard of size n x n, where n is a power of 2, with L-shaped tiles (trominoes), except for one defective square. The algorithm divides the board recursively into equal half-sized subboards until reaching boards of size 2x2, which can be covered with at most one tile. It then combines the solutions to the subproblems to construct a solution for the original board. The time complexity of this divide and conquer algorithm is O(n^2).
Rational equations
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.
What's hot (20)
3m. Pedagogy of Mathematics (Part II) - Algebra (Ex 3.13)
3m. Pedagogy of Mathematics (Part II) - Algebra (Ex 3.13)
Similar to Factoring Pt. 2/2 (x^2 + bx + c)
Factor by grouping
The document provides steps for factoring polynomial expressions:
1) Separate the expression into terms with common factors and group those terms;
2) Find the greatest common factor (GCF) of the grouped terms;
3) Enclose the GCF and the grouped terms in parentheses to arrive at the factored expression.
Worked examples demonstrate factoring various polynomials, including using negative signs properly.
Polynomial math
1) The document discusses various methods for manipulating and solving algebraic expressions, including adding, subtracting, and factoring polynomials.
2) Factoring techniques include grouping like terms, using the difference of squares formula, and recognizing perfect square trinomials.
3) The quadratic formula is introduced as a way to solve quadratic equations of the form ax2 + bx + c = 0.
Chapter 2.5
The document discusses the distributive property and combining like terms in algebra. It defines key terms such as terms, coefficients, and like terms. It then explains the distributive property using examples of distributing a number over terms in parentheses. Finally, it provides practice problems for students to work through using the distributive property to combine like terms.
Completing the square
1) The document discusses completing the square, which involves transforming quadratic expressions into perfect square trinomial form.
2) An example problem walks through completing the square to solve the equation x2 + 6x = 7. This involves adding a constant term, finding the constant to make the expression a perfect square, and then using square roots to solve for x.
3) Another example solves the equation x2 + 10x - 8 = 0 by following similar steps but leaving the radical term in the solution instead of simplifying further.
Feb 18 Dividing Polynomials By Binomials 2
This document discusses dividing polynomials by binomials. It provides examples of polynomials that can and cannot be divided by binomials. Specifically, it shows that x^2 + 5x + 7 cannot be factored, but can be divided by (x+2), yielding an "ugly" quotient of (x+3) + 1/(x+2). It also notes that complicated polynomials that are hard to simplify can still be divided by binomials, again yielding an ugly quotient. Further examples provided include dividing 4x^3 - 1 + 8x by 4 + 4x and 6y^3 - 4y^2 - 9y - 3 by 2y^2 - 3. The document concludes with assigning exercises 1
Factoring Special Products in Difference of Squares
The document provides examples of factoring quadratic expressions of the form a^2 +/- b^2 into the form (a +/- b)(a -/+ b). It demonstrates factoring several quadratic expressions using this method, including expressions with coefficients other than 1. It also introduces a formula for factoring expressions of the form a^2 + 2ab + b^2 and applies this formula to one example. Finally, it factors one expression where the middle term is negative by distributing the negative sign appropriately.
Solving polynomial equations in factored form
The document provides instructions for solving polynomial equations in factored form. It begins by explaining that to solve an equation like (x – 5)(x + 4) = 0, one should not use the FOIL method but rather split the equation into two separate problems that each equal zero: x – 5 = 0 and x + 4 = 0. It then works through several examples of solving factored polynomial equations by finding the values of x that make each factor equal to zero. The document also covers factoring out the greatest common factor from expressions.
Factoring quadratic expressions
1) There are several methods for solving quadratic equations, including factoring, graphing, using the quadratic formula, and completing the square.
2) Factoring involves expressing the quadratic expression as a product of two linear factors. Methods for factoring include finding the greatest common factor, using factor diamonds, grouping, and the borrowing method.
3) The document provides examples of solving quadratics using various factoring techniques and practicing additional problems.
Factoring Trinomials
1. The document discusses various methods for factoring trinomials, including using algebra tiles and looking for pairs of numbers that multiply to the constant term and add to the coefficient of the middle term.
2. It provides examples of factoring different types of trinomials step-by-step, including those with and without leading coefficients.
3. Some trinomials cannot be factored, like those where the pairs of numbers do not add up to the middle coefficient, making the expression prime.
Lecture 03 special products and factoring
This document provides information and examples on multiplying polynomials, including:
1) Multiplying a monomial and polynomial using the distributive property.
2) Multiplying two polynomials using both the horizontal and vertical methods.
3) Factoring trinomials and identifying similar and conjugate binomials. Methods like FOIL and grouping are discussed.
Polynomials Grade 10
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.
Factoring and Box Method
This document discusses various techniques for factoring polynomials, including:
1. Factoring using the greatest common factor (GCF).
2. Factoring polynomials with 4 or more terms by grouping.
3. Factoring trinomials using factors that add up to the coefficient of the middle term.
4. Using the "box method" to factor trinomials where the coefficient of the x^2 term is not 1.
5. Factoring the difference of two squares using the formula a^2 - b^2 = (a + b)(a - b).
sim-140907230908-phpapp01.pptx
This document provides an overview of solving quadratic equations by factoring. It discusses identifying quadratic equations, rewriting them in standard form, factoring trinomials in the form x^2 + bx + c, and determining roots. Several examples of factoring trinomials and solving quadratic equations are shown. Activities include identifying quadratic equations, rewriting equations in standard form, factoring trinomials, and solving equations by factoring. The document provides resources for further learning about quadratic equations and factoring.
Solving Quadratic Equations by Graphing
This document provides instructions for solving quadratic equations by graphing. It begins with an overview of standard form and then works through two example problems. For problem 1, it graphs the equation x^2 + 4x - 5 = 0 and finds the vertex, y-intercept, line of symmetry, and x-intercepts to check the solution. For problem 2, it similarly graphs x^2 + 4x = -6 and finds the relevant points. It concludes that if a parabola does not intersect the x-axis, it represents no solution to the equation.
Quadratic Inequalities.pptx
The document provides 16 quadratic equations and factorizes them to solve for the values of x that satisfy each equation. It then provides 12 additional quadratic expressions and solves the corresponding inequalities to determine the range of values for x where each expression is positive, negative, or less than/greater than zero.
Polynomials review
The document discusses multiplying polynomials using the box method. It introduces the box method as analogous to a Punnett square from genetics. The steps of the box method are: 1) place the terms in the first row and column, 2) multiply corresponding terms, 3) combine like terms, and 4) write in standard form. An example of (z - 5)(z + 3) is worked out following these steps. The document concludes with notes on multiplying polynomials using the box method and a practice problem.
Rational Expressions
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.
March 8
1. The document provides notes and assignments for a math class. It summarizes topics covered in class including factoring perfect square trinomials and special case binomials.
2. Upcoming assignments include a Khan Academy assignment due Sunday and a test on factoring trinomials and binomials on Tuesday. Late work policies are outlined.
3. Sample problems are worked through for factoring perfect square trinomials and special case binomials like the difference and sum of squares. Students are to complete similar practice problems for homework.
0408 ch 4 day 8
The document provides steps to solve exponential and logarithmic equations:
1. Isolate the exponential expression.
2. Take the log of both sides.
3. Solve and verify all solutions by substitution.
It then works through examples of solving exponential equations, isolating the exponential term, taking logs of both sides, and solving for the variable. Solutions are verified with substitution.
Similar to Factoring Pt. 2/2 (x^2 + bx + c) (20)
Factoring Special Products in Difference of Squares
Factoring Special Products in Difference of Squares
More from ListeningDaisy
Geometry dots & lines
This document provides an overview of basic geometry concepts including:
- Points have no dimension and are named using capital letters
- A line has no end points, is straight, and goes off to infinity in two directions
- A line segment is part of a straight line with two distinct end points
- A ray has one endpoint and extends off to infinity in one direction
- Lines, line segments, and rays can intersect at a single point
- A plane is a flat, two-dimensional surface that can be named using points or a cursive letter
Stem & leaf plots + histograms
This document provides instructions and examples for creating different types of data displays, including stem-and-leaf plots, histograms, back-to-back stem-and-leaf plots, and box-and-whisker plots. It includes examples using temperature data and adoption prices for dogs. Key terms and how to interpret the visuals are explained through mini lessons within the document.
Measures of central tendency
This document discusses measures of central tendency and uses the scores of 28, 18, 24, 22, 18 to demonstrate how to calculate the median, mean, mode, and range. It defines median as the middle number when values are ordered from lowest to highest. Mean is calculated by adding all values and dividing by the total count. Mode is the number with the most occurrences. Range is found by subtracting the minimum from the maximum value.
Factoring polynomials completely
The document provides instructions on how to completely factor polynomials. It gives examples of factoring expressions like 4n^2 - 16, 5x^2 - 45, and 3x^2 - 18x - 21. The steps explained are to find the greatest common factor, factor any binomials or trinomials, and simplify the expression as much as possible. Additional practice problems and their factored solutions are provided.
GCF & LCM
This document discusses the Green Climate Fund (GCF), an international organization that aims to support climate change mitigation and adaptation projects in developing countries. The GCF provides funding to projects that help reduce greenhouse gas emissions and make developing nations more resilient to climate change impacts. The entire document focuses on describing the goals and work of the Green Climate Fund.
Multiplying Polynomials
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive functioning. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms.
Math vocab 1
1. The document defines key math vocabulary terms including variable, constant, coefficient, leading coefficient, like terms, monomial, polynomial, binomial, and trinomial.
2. Examples are provided to demonstrate identifying variables, constants, coefficients, like terms, and determining if expressions are monomials, polynomials, binomials, or trinomials.
3. The degree of a polynomial is defined as the highest exponent present in the polynomial. Examples show determining the degree of various polynomials.
More from ListeningDaisy (7)
Factoring Pt. 2/2 (x^2 + bx + c)
- 1. Factoring Pt. 2/2 (x2 + bx + c) By L.D.
- 2. Problem 1 3x2 + 8x + 4
- 3. Problem 1 3x2 + 8x + 4 The first thing we have to do is to find what multiplies to get 3x2. I choose 3x and x, so we put them in parenthesis; (3x + ?)(x + ?)
- 4. Problem 1 3x2 + 8x + 4 (3x + ?)(x + ?) The next step is a bit different from what you are expecting, we will take multiplication problems that make 4 and place them in the question mark’s spots, forgetting about the 8 for now. You will do will differentiate between the problems like 1(4) and 4(1) giving them separate problems. These will be made and solved on the next page. Pairs that make 4: 2 and 2, 1 and 4, 4 and 1
- 5. Problem 1 1. (3x + 2)(x + 2) 2. (3x + 1)(x + 4) 3. (3x + 4)(x + 1) Use F.O.I.L. to solve these…
- 6. Problem 1 1. (3x + 2)(x + 2) 3x2 + 4 + 8x 2. (3x + 1)(x + 4) 3x2 + 4 + 4x 3. (3x + 4)(x + 1) 3x2 + 4 + 7x We will now remember the 8 and our original problem (3x2 + 8x + 4), since the top problem contains an 8 and is exactly the same to our problem, our answer is (3x + 2)(x + 2).
- 7. Problem 2 2x2 + 13h - 7
- 8. Problem 2 2x2 + 13h – 7 1. For our first step, I will say that to make 2x you need 2x and x. 2. Going to our second step we will say that to get -7 you can use -1 and 7, 1 and -7, 7 and -1, -7 and 1 (2x + -1) (x + 7) (2x + 1) (x + -7) (2x + 7) (x +-1) (2x + -7) (x + 1)
- 9. Problem 2 2x2 + 13h – 7 (2x + -1) (x + 7) 2x2 – 7 + 13x (2x + 1) (x + -7) 2x2 – 7 + -13x (2x + 7) (x +-1) 2x2 – 7 + 5x (2x + -7) (x + 1) 2x2 – 7 – 5x
- 10. Mini Lesson Do you remember how in the last slide, when we had problems like (2x + 7) (x +-1) & (2x + -7) (x + 1) where the negative sign just switched sides? A cheat to answer that is to write answer of the problem for the first one (2x2 – 7 + 5x) and when you are answering the next one you can just add a negative to the coefficient.
- 11. Problem 2 (2x + -1) (x + 7) is the answer since it multiplies to make our original problem, 2x2 + 13h – 7.
- 12. Problem 3 -x2 + x + 20
- 13. Problem 3 -x2 + x + 20 The way to get around the negative sign on the first problem is to change it to -1(x2 – x – 20) and solve the problem inside the parenthesis ignoring the -1. -1(x – 5)(x + 4) Now you can multiply it all by -1. (-x + 5)(-x - 4) This is your final answer.