5007 Expert Voices Jered Bright
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The document discusses four methods for solving monic quadratic equations: square roots, factoring, completing the square, and the quadratic formula. It provides examples of how to use each method, including isolating terms, adding or subtracting quantities, taking square roots, and factoring expressions. The quadratic formula is presented as a reliable method that can always be used to solve a quadratic equation.
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True or False: x=14 is a solution to x+8=23. True or False: x=14 is a solution to x-10=4. Evaluate the expression x^3-5x for x= -2. If a state’s income tax is 4.3% of taxable income, how much will be owed in tax if your taxable income is $51,000? Solve the following equation: X+2.29=8.27.
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Rational numbers are numbers that can be written as fractions p/q, where p and q are integers and q is not equal to 0. Rational numbers have important properties:
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This document discusses square roots and irrational numbers. It defines that the square of an integer is a perfect square, and taking the square root is the opposite of squaring a number. Examples are provided of finding the number that produces a square of 81, which is 9, but -9 also works. The document also defines rational and irrational numbers, noting that square roots of perfect squares are rational, while square roots of non-perfect squares are irrational.
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1) Standard form for writing equations of lines is Ax + By = C, where you can find A, B, and C using two points on the line.
2) Slope-intercept form is y = mx + b, where m is the slope found using two points and b is the y-intercept.
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- A linear inequality describes a region of the coordinate plane bounded by a line. Any point in the shaded region is a solution to the inequality.
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3) To find the equation of a slant asymptote, divide the numerator by the denominator and keep only the terms with the highest powers of x.
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The Bairstow method uses synthetic division to iteratively calculate the roots of a polynomial by dividing it by quadratic factors (x^2 - rx - s) until the remainder is zero. It obtains better approximations of r and s at each iteration to isolate the roots.
The Muller method fits a parabola through three initial guesses to obtain coefficients a, b, c. It then uses the quadratic formula on the parabola to find the root, and iterates with new guesses until converging on a solution.
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This document reviews representations of different types of numbers on the number line. It discusses natural numbers, integers, rational numbers like terminating and repeating decimals. Irrational numbers like √2 that are non-terminating and non-repeating are also reviewed. Key properties of real numbers are listed, including that every point on the number line corresponds to a unique real number, and real numbers satisfy closure, commutative, associative, identity and inverse properties.
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This document discusses different types of real numbers including rational and irrational numbers. It defines rational numbers as numbers that can be expressed as p/q where p and q are integers. Irrational numbers are defined as numbers with a non-terminating and non-repeating decimal representation. Euclid's division lemma and algorithm are explained as ways to find the highest common factor of two numbers. The fundamental theorem of arithmetic states that every composite number can be expressed as a product of primes. Relationships between the highest common factor and lowest common multiple of numbers are also covered. Finally, the document defines terminating, repeating, and non-terminating decimals.
CrespoSampleof Student Correspondence
Megan provides explanations for several math problems from examples in the textbook. For problem 5-2 #2a, Megan explains that the diagram shows segment KH has a length of 10, so segment KD must also be 10 based on similar markings. For problem 5-3 #1a, Megan lists the steps to find the center of a circle circumscribing a triangle by finding the perpendicular bisectors of two sides and their point of intersection. For problem 5-3 #1b, Megan notes that finding a third perpendicular bisector is not necessary since the first two already intersect at the third side. For problem 5-3 #3, Megan shows that the total length can be broken into parts
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This document discusses square roots and irrational numbers. It defines that the square of an integer is a perfect square, and taking the square root is the opposite of squaring a number. Examples are provided of finding the number that produces a square of 81, which is 9, but -9 also works. The document also defines rational and irrational numbers, noting that square roots of perfect squares are rational, while square roots of non-perfect squares are irrational.
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1) Standard form for writing equations of lines is Ax + By = C, where you can find A, B, and C using two points on the line.
2) Slope-intercept form is y = mx + b, where m is the slope found using two points and b is the y-intercept.
3) The slope of a line perpendicular to another is the opposite reciprocal of the original line's slope. You can find the perpendicular equation using the point-slope formula.
Graphing Linear Inequalities
- A linear inequality describes a region of the coordinate plane bounded by a line. Any point in the shaded region is a solution to the inequality.
- To graph a linear inequality, first solve it for y and graph the resulting equation as a line. Then, test a point not on the line to determine which side of the line to shade based on whether it satisfies the inequality.
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Slope-intercept form is represented by the equation y=mx+b, where m is the slope and b is the y-intercept. There are three methods to find the slope: rise over run, change in y over change in x, and point-slope formula. The slope tells the rate of change, while the y-intercept is the initial value when x=0. To graph a line in slope-intercept form, you plot the y-intercept and then use the slope to trace successive points on the line by calculating the rise over run at each interval of x. Word problems can be modeled using this form by identifying the independent and dependent variables and finding their relationship through calculating slope from ordered pairs of known values.
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Slope-intercept form is represented by the equation y=mx+b, where m is the slope and b is the y-intercept. There are three methods to find the slope: rise over run, change in y over change in x, and point-slope formula. The slope tells the rate of change, while the y-intercept is the initial value when x=0. To graph a line in slope-intercept form, you plot the y-intercept and then use the slope to trace successive points on the line by calculating the rise over run at each interval. Word problems can be modeled using this form by identifying the independent and dependent variables and finding their relationship through calculating slope from ordered pairs of known values.
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Irrational numbers are any numbers that cannot be expressed as a quotient of two integers. Examples of irrational numbers include the square root of 2 and the square root of 30. To determine if a principal root is rational or irrational, check if the radicand is a perfect nth power - if it is, the root is rational, otherwise it is irrational. Estimating square roots involves finding the two closest perfect squares that are less than and greater than the number inside the radical.
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This document discusses rational functions and their graphs. It outlines 3 cases for rational functions based on the degree of the numerator and denominator. It also discusses identifying and plotting asymptotes, including vertical, horizontal, and slant asymptotes. Methods are provided for graphing rational functions based on finding asymptotes and plotting points around the asymptotes to determine the graph's behavior. Examples are worked through demonstrating these graphing techniques.
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This document discusses rational functions and their properties. It defines a rational function as a function of the form f(x) = p(x)/q(x) where p and q are polynomials. It then discusses how to find the domain, vertical asymptotes, horizontal asymptotes, and oblique asymptotes of rational functions. The key points are: 1) the domain excludes values where the denominator equals 0, 2) vertical asymptotes occur where the denominator equals 0, 3) the degree of the numerator and denominator determine if there is a horizontal or oblique asymptote. Comparing degrees is essential to finding asymptotes of rational functions.
Asymptotes and holes 97
1) Points of discontinuity occur when the denominator of a rational function equals zero.
2) A rational function can have at most one horizontal asymptote. If the degree of the numerator is less than the denominator, the horizontal asymptote is y=0. If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients.
3) To find the equation of a slant asymptote, divide the numerator by the denominator and keep only the terms with the highest powers of x.
Roots of polynomials
The Bairstow method and Muller method are techniques for calculating the roots of polynomials.
The Bairstow method uses synthetic division to iteratively calculate the roots of a polynomial by dividing it by quadratic factors (x^2 - rx - s) until the remainder is zero. It obtains better approximations of r and s at each iteration to isolate the roots.
The Muller method fits a parabola through three initial guesses to obtain coefficients a, b, c. It then uses the quadratic formula on the parabola to find the root, and iterates with new guesses until converging on a solution.
Both methods use iterative techniques to refine initial guesses and isolate the roots of polynomials without using
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Real numbers include all rational and irrational numbers. There are several key properties and theorems regarding real numbers:
1) Euclid's division lemma states that for any two positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b. This lemma is used to calculate the highest common factor of two integers.
2) The fundamental theorem of arithmetic states that every integer greater than 1 can be written as a unique product of prime factors, regardless of their order.
3) Rational numbers are numbers that can be written as a ratio of two integers. Irrational numbers are all other real numbers that are not rational and have non-repe
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The document reviews slope and the slope-intercept formula y=mx+b. It provides examples of finding the slope and y-intercept of various lines given their graphs or two points on the line. It also demonstrates how to write the equation of a line given two points and how to find the x- and y-intercepts of a line given its equation.
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The document discusses point-slope form and writing linear equations. It provides examples of writing equations for lines that are parallel and perpendicular to given lines. Specifically, it explains that parallel lines have the same slope and different y-intercepts, while perpendicular lines have slopes that are negative reciprocals of each other. Examples are given of writing equations for parallel and perpendicular lines using point-slope form based on a given point and the slope of another line.
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This document discusses the history and key concepts of real numbers. It provides background on how real numbers developed from ancient civilizations working with simple fractions to the formal acceptance of irrational numbers. Key figures discussed include Euclid, Hippasus, and developments in ancient Egypt, India, Greece, the Middle Ages, and Alexandria. Fundamental ideas covered include Euclid's lemma, the fundamental theorem of arithmetic, prime factorisation, and the distinction between rational and irrational numbers.
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Advance algebra
This document provides information about several advanced algebra topics:
1. The Remainder Theorem states that when dividing a polynomial f(x) by x-c, the remainder is equal to f(c).
2. The Factor Theorem states that if f(c) = 0, then x-c is a factor of the polynomial.
3. Synthetic division is a method for dividing polynomials without subtracting, by multiplying the coefficients.
4. The Rational Zeros Theorem can be used to find all possible rational zeros of a polynomial by considering factors of the leading coefficient and constant term.
Solving cubic equations with the help of factor theorem
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The coefficients of a, b, c and d are real or complex numbers with a not equals to zero (a ≠ 0). It must have the term x3 in it, or else it will not be a cubic equation. But any or all of b, c and d can be zero.
Maths notes for 4038 and 4016 paper
This document provides tips for acing the Additional Mathematics (AM) and Elementary Mathematics (EM) exams. It summarizes key statistics on topics that are highly tested, such as differentiation and integration making up 27.8% of the AM exam. It recommends focusing on the 11 chapters that make up 74.6% of the exam. Sample questions are provided for topics like trigonometry, logarithms, linear laws, and matrices. Strategies are outlined for solving different types of questions on these topics.
The Many Forms of Quadratic Equations
The document discusses three forms of quadratic equations - standard form, vertex form, and intercept form. It provides the key characteristics of each form and steps for graphing equations written in each form. Standard form is easiest to solve by factoring. Vertex form shows the vertex and axis of symmetry. Intercept form directly lists the x-intercepts. Converting between forms involves multiplying out terms.
Single Variable Calculus Assignment Help
I am Duncan V. I am a Single Variable Calculus Assignment Solver at mathhomeworksolver.com. I hold a Master's in Mathematics, from Manchester, United Kingdom. I have been helping students with their assignments for the past 12 years. I solved assignments related to Single Variable Calculus.
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Rs solving graphingquadraticequation
The document discusses quadratic equations and functions. It defines a quadratic equation as one where the variable is squared, and explains that the graph of a quadratic function is a parabola with an open U-shape. It outlines the standard form of a quadratic equation as ax2 + bx + c = 0 and describes how to find the solutions/roots by setting it equal to zero. Methods for solving quadratic equations include using the quadratic formula, factoring, or completing the square. The key steps to graph a quadratic equation are finding the standard form, solutions/roots, and y-intercept.
Quadratic functions my maths presentation
1. The document discusses key concepts related to quadratic equations including factorizing, solving, graphing, finding roots and axis of symmetry, locating maximum/minimum values, and using graphs to solve related equations and inequalities.
2. The key steps provided to factorize a quadratic equation are to find the product and factors of the coefficients that add up to the middle term, and then pair the factors.
3. To solve a quadratic equation, take all terms to one side, factorize, set each factor equal to 0 and solve for x to find the two roots.
Single Variable Calculus Assignment Help
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Single Variable Calculus Assignment Help
I am Arcady N. I am a Single Variable Calculus Assignment Expert at mathsassignmenthelp.com. I hold a Master's in Mathematics from, Queen’s University. I have been helping students with their assignments for the past 12 years. I solve assignments related to Single Variable Calculus.
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0010 chapter iii
Here are the steps to solve this quadratic equation using the quadratic formula:
1. Identify the coefficients: a = 3, b = -1, c = -5/2
2. Plug the coefficients into the quadratic formula:
x = (-b ± √(b2 - 4ac)) / 2a
x = (-(-1) ± √((-1)2 - 4(3)(-5/2))) / 2(3)
3. Simplify:
x = (1 ± √(1 + 30)) / 6
4. Evaluate the square root:
x = (1 ± 5√3) / 6
Therefore, the solutions are:
Completing the square (added and revised)
The document discusses solving quadratic equations by completing the square. It provides examples of perfect square trinomials and explains the steps to solve quadratic equations using completing the square. These steps include: 1) moving terms to one side of the equation, 2) finding the term to complete the square and adding it to both sides, 3) factoring the perfect square trinomial, 4) taking the square root of both sides, and 5) solving for the two possible solutions. It also provides a second example where the quadratic coefficient is not 1, requiring dividing all terms by this coefficient first before completing the square.
Tricks to remember the quadratic equation.ACTION RESEARCH ON MATHS
This document provides information about different methods for solving quadratic equations. It discusses factoring the equation, using the quadratic formula, and completing the square. Step-by-step explanations are provided for each method. Factoring involves finding two binomials that multiply to give the quadratic term and add to the linear term. The quadratic formula is given as x = (-b ±√(b2 - 4ac))/2a. Completing the square requires grouping like terms and completing the square of the quadratic term.
Roots of polynomials
The Bairstow method and Muller method are techniques for calculating the roots of polynomials.
The Bairstow method uses synthetic division to iteratively calculate the roots of a polynomial by dividing it by quadratic factors (x^2 - rx - s) until the remainder is zero. It obtains better approximations of r and s at each iteration to isolate the roots.
The Muller method fits a parabola through three initial guesses to obtain the coefficients a, b, c of the quadratic formula. It then uses the quadratic formula to find the root, and iterates with new guesses to converge on more accurate roots.
Both methods use iterative techniques to gradually converge on the real and complex roots of polynomials without
Quadractic equations.steps
The document explains how to use the quadratic formula to find the roots or solutions of a quadratic equation. It provides the steps which are: 1) write the equation in standard form with all terms on one side, 2) identify the coefficients a, b, and c, and 3) substitute these coefficients into the quadratic formula. The formula is given as x = (-b ±√(b2 - 4ac))/2a. Worked examples are provided to demonstrate how to set up and solve a quadratic equation using the formula.
Tutorial linear equations and linear inequalities
This document discusses linear equations and inequalities in one variable. It begins by defining open sentences, variables, and solutions. It then covers topics like solving linear equations using addition, subtraction, multiplication, and division. It also discusses solving multi-step equations. Graphing solutions to equations is explained. The document also covers understanding and solving linear inequalities in one variable as well as graphing inequalities. It provides examples of how equations and inequalities can be applied to everyday situations.
Review packet questions
This document provides instructions for solving several calculus problems involving derivatives. It explains how to take the derivative of functions using power rules, product rules, chain rules and implicit differentiation. It also provides guidance on using derivatives to find points where the tangent line has a particular slope or is horizontal. Finally, it demonstrates how to apply the quotient rule when given function values rather than the actual functions.
quadraticequations-111211090004-phpapp02.pptx
Quadratic equations are polynomial equations of the second degree that take the general form of ax2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. They can be solved using methods like factoring, completing the square, or the quadratic formula. The discriminant, b2 - 4ac, determines whether the roots are real numbers, a repeated real root, or complex numbers.
Theory of Equation
1. The document provides an overview of the key concepts in the Theory of Equations unit, including types of equations, methods for solving different types of equations, and properties of roots.
2. It discusses linear equations, simultaneous equations, quadratic equations and their solving methods like elimination, substitution, and factorization.
3. Examples of equation problems from commercial applications are also presented, involving linear, simultaneous and quadratic equations. Worked examples and practice problems are provided for each topic.
Math Project 2ppt
This document provides an overview of solving different types of equations and inequalities, including linear equations, quadratic equations, rational equations, radical equations, linear inequalities, quadratic inequalities, and graphing methods. Key methods discussed are factoring, the quadratic formula, completing the square, manipulating fractions to get a common denominator, squaring both sides of an equation to eliminate radicals, and determining shading regions for inequalities based on a test point.
.
This document provides information about solving absolute value equations and inequalities, as well as quadratic equations. It discusses:
1) To solve absolute value equations, you must divide the equation into two equations by treating the expression inside the absolute value bars as both positive and negative.
2) For inequalities, the direction of the inequality sign must be reversed when multiplying or dividing both sides by a negative number.
3) Quadratic equations can be solved by factoring if possible, or using the quadratic formula. The discriminant determines the number of real roots.
Similar to 5007 Expert Voices Jered Bright (20)
Solving cubic equations with the help of factor theorem
Solving cubic equations with the help of factor theorem
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After this second you should be able to: Explain the main determinants of demand for any mineral product, and their relative importance; recognise and explain how demand for any product is likely to change with economic activity; recognise and explain the roles of technology and relative prices in influencing demand; be able to explain the differences between the rates of growth of demand for different products.
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BONKMILLON Unleashes Its Bonkers Potential on Solana.pdf
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Let's be real for a second – the world of meme coins can feel like a bit of a circus at times. Every other day, there's a new token promising to take you "to the moon" or offering some groundbreaking utility that'll change the game forever. But how many of them actually deliver on that hype?
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OJP data from firms like Vicinity Jobs have emerged as a complement to traditional sources of labour demand data, such as the Job Vacancy and Wages Survey (JVWS). Ibrahim Abuallail, PhD Candidate, University of Ottawa, presented research relating to bias in OJPs and a proposed approach to effectively adjust OJP data to complement existing official data (such as from the JVWS) and improve the measurement of labour demand.STREETONOMICS: Exploring the Uncharted Territories of Informal Markets throug...
Delve into the world of STREETONOMICS, where a team of 7 enthusiasts embarks on a journey to understand unorganized markets. By engaging with a coffee street vendor and crafting questionnaires, this project uncovers valuable insights into consumer behavior and market dynamics in informal settings."
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我们从事工作十余年的有着丰富经验的业务顾问，熟悉海外各国大学的学制及教育体系，并且以挂科生解决毕业材料不全问题为基础，为客户量身定制1对1方案，未能毕业的回国留学生成功搭建回国顺利发展所需的桥梁。我们一直努力以高品质的教育为起点，以诚信、专业、高效、创新作为一切的行动宗旨，始终把“诚信为主、质量为本、客户第一”作为我们全部工作的出发点和归宿点。同时为海内外留学生提供大学毕业证购买、补办成绩单及各类分数修改等服务；归国认证方面，提供《留信网入库》申请、《国外学历学位认证》申请以及真实学籍办理等服务，帮助众多莘莘学子实现了一个又一个梦想。
专业服务，请勿犹豫联系我
如果您真实毕业回国，对于学历认证无从下手，请联系我，我们免费帮您递交
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你赢我赢，共创双赢
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Detailed power point presentation on compound interest and how it is calculated
Detailed information about compund interest
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This blog explores the current largest producer of soybean in India and the factors contributing to their dominance.
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"Does Foreign Direct Investment Negatively Affect Preservation of Culture in the Global South? Case Studies in Thailand and Cambodia."
Do elements of globalization, such as Foreign Direct Investment (FDI), negatively affect the ability of countries in the Global South to preserve their culture? This research aims to answer this question by employing a cross-sectional comparative case study analysis utilizing methods of difference. Thailand and Cambodia are compared as they are in the same region and have a similar culture. The metric of difference between Thailand and Cambodia is their ability to preserve their culture. This ability is operationalized by their respective attitudes towards FDI; Thailand imposes stringent regulations and limitations on FDI while Cambodia does not hesitate to accept most FDI and imposes fewer limitations. The evidence from this study suggests that FDI from globally influential countries with high gross domestic products (GDPs) (e.g. China, U.S.) challenges the ability of countries with lower GDPs (e.g. Cambodia) to protect their culture. Furthermore, the ability, or lack thereof, of the receiving countries to protect their culture is amplified by the existence and implementation of restrictive FDI policies imposed by their governments.
My study abroad in Bali, Indonesia, inspired this research topic as I noticed how globalization is changing the culture of its people. I learned their language and way of life which helped me understand the beauty and importance of cultural preservation. I believe we could all benefit from learning new perspectives as they could help us ideate solutions to contemporary issues and empathize with others.快速制作美国迈阿密大学牛津分校毕业证文凭证书英文原版一模一样
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5007 Expert Voices Jered Bright
- 1. Solving (Monic) Quadratic Equations The quick and dirty methods you may have learned simplified!
- 2. In solving quadratic equations, there are three ways in which to solve for x: - by square roots (directly) - by factoring - by completing the square - by the quadratic formula **Note: in this presentation, we assume all quadratic equations are monic, meaning . If this is not the case, divide all terms by “a” to make it monic.
- 4. Square root both sides!
- 6. -by square roots -by factoring -by completing the square -by the quadratic formula Isolate the x term….
- 7. -by square roots -by factoring -by completing the square -by the quadratic formula Square root both sides….
- 8. -by square roots -by factoring -by completing the square -by the quadratic formula Remember the plus and minus….
- 10. With the numbers from the first step, determine if any combination of the numbers when added together result in “b”.
- 11. These numbers from the second step (if it works) now need to be written as (x-(first number) )x(x-(second number))=0
- 13. -by square roots -by factoring -by completing the square -by the quadratic formula Determine the numbers that multiply together to be equal to -6:
- 14. -by square roots -by factoring -by completing the square -by the quadratic formula Now look at the sum of each of these pairs of numbers. If one pair is equal to -1=b, those are the numbers used to factor the expression
- 15. -by square roots -by factoring -by completing the square -by the quadratic formula Having identified our factors, we can write the above expression as….
- 16. -by square roots -by factoring -by completing the square -by the quadratic formula Lastly, we set each part (factor) equal to zero and solve for x:
- 18. Identify the “bx” term in your equation and add to both sides of the equation.
- 19. After this point, the side containing the x terms will factor as
- 21. -by square roots -by factoring -by completing the square -by the quadratic formula First we move the x terms to one side of the equation and the other terms to the other side of the equation….
- 22. -by square roots -by factoring -by completing the square -by the quadratic formula Next, we determine our b value and compute And add it to each side of the equation. So,
- 23. -by square roots -by factoring -by completing the square -by the quadratic formula After this, the left side of the equation will factor (this is left up to you to verify) as
- 24. -by square roots -by factoring -by completing the square -by the quadratic formula Lastly, finish solving for x using square roots and other basic algebra skills:
- 26. Substitute in the appropriate values into the following formula and simplify to obtain your solutions:(This is the quadratic formula)
- 27. -by square roots -by factoring -by completing the square -by the quadratic formula This problem can be solved by completing the square, but the quadratic formula is going to be used to determine the x values.
- 28. -by square roots -by factoring -by completing the square -by the quadratic formula First we determine our a, b, and c:
- 29. -by square roots -by factoring -by completing the square -by the quadratic formula Then we use the quadratic formula with these values: Thus