Worried about due completion of math homework? You can count on us. In Homework1.com we have excellent infrastructure to serve you even at the most critical hours of assignment submission! Try our service today and get excellent score in math exam.
This document discusses exponent rules and formulas involving positive, negative, fractional and zero exponents. It provides examples of simplifying expressions using these rules. Key points covered include:
- Formula I: aman = am+n
- Formulas II-V cover properties for negative exponents and fractional exponents
- Examples are worked through applying the exponent rules and formulas
The document provides 10 examples of word problems involving quadratic equations. For each problem it defines the variables, sets up the quadratic equation, solves for the zeros, and states the answer.
The document discusses solving quadratic equations. It provides examples of solving quadratic equations by factoring, completing the square, and using the quadratic formula. Various techniques are demonstrated including finding the solutions sets for quadratic equations.
This document provides examples and explanations for multiplying, dividing, and otherwise working with rational expressions. It defines rational expressions as involving polynomials in both the numerator and denominator, with the denominator not equal to 0. Examples are provided for multiplying and dividing rational expressions involving monomials and polynomials. A multi-step word problem is also worked through to find a model for and approximate a baseball player's career batting average as a rational expression involving years played.
The document provides instructions for graphing and solving various types of quadratic equations. It defines standard form, vertex form, and intercept form of quadratics. It explains how to graph quadratics by finding the vertex and intercepts. Methods covered include factoring, taking square roots, completing the square, and using the quadratic formula. Examples are included to demonstrate each process.
CAT Quadratic and Higher Order EquationsGeorge Prep
油
This document discusses quadratic and higher order equations. It begins by introducing polynomials and classifying them based on coefficients and degree. Quadratic equations are then examined in more detail, including forming quadratic equations, finding roots using methods like the quadratic formula, and solving quadratic inequalities. The document also covers topics like the sum and product of roots, the discriminant and nature of roots, maximum/minimum values, common roots between equations, and relationships between coefficients and roots for higher order equations. Descartes' rule of signs is presented as a way to determine the number of positive and negative real roots of a polynomial equation.
This document provides information about an assignment for a math course. It lists 27 problems related to systems of equations and matrices that students need to complete by the due date of July 20, 2014. It provides instructions on entering answers in WeBWorK and notes functions it can understand. It encourages students to ask for help if struggling rather than guessing answers.
This document provides instruction on solving algebraic equations that have variables on both sides. It begins with a review of solving equations with a variable on one side, such as 6x+4=28. It then demonstrates how to solve equations with variables on both sides through a step-by-step process of combining like terms, moving terms to one side of the equation, and then dividing both sides by the coefficient of the variable. Several examples are worked through and solutions are checked by substituting the solutions back into the original equations. The document concludes by providing additional practice problems for the student to solve.
This document discusses solving systems of equations by elimination. It provides examples of eliminating a variable by adding or subtracting equations. The key steps are: 1) write the equations in standard form; 2) add or subtract the equations to eliminate one variable; 3) substitute the eliminated variable back into one equation to solve for the other variable. Checking the solution in both original equations verifies the correct solution was found.
This document provides solutions to exercises from NCERT Class 9 Maths Chapter 4 on linear equations in two variables. It includes:
1) Solving linear equations representing word problems and expressing equations in the form ax + by + c = 0.
2) Finding solutions that satisfy given linear equations and determining the value of k if a given point is a solution.
3) Drawing graphs of various linear equations by plotting points that satisfy each equation.
4) Giving two equations of lines passing through a point and noting there are infinitely many such lines.
This document contains an exercise set with 46 problems involving real numbers, intervals, and inequalities. The problems cover topics such as determining whether numbers are rational or irrational, solving equations, graphing inequalities on number lines, factoring polynomials, and solving compound inequalities.
1. The document provides 10 problems to calculate the derivative (dy/dx) of various functions involving radicals, trigonometric functions, and composition of functions.
2. The solutions show setting the radical or function within a radical equal to a variable u, then calculating du/dx and applying the chain rule and power rule to determine dy/dx.
3. The answers are provided in fractional form with radicals, involving variables and parameters from the original functions.
This document provides information about an online math homework help service called Homework1. It includes:
- Contact information for Homework1, including their address, phone number, email, and social media links.
- An overview of the services provided, which include math homework help, writing assistance, and teaching students the solutions to help them learn.
- Several examples and solutions to common math problems to illustrate the type of homework help offered.
This document discusses how to solve radical equations by isolating the radical expression, removing the radical sign by raising both sides to the appropriate power, and solving the resulting equation. It provides examples of solving various radical equations step-by-step and checking solutions. Key steps include isolating the radical term, removing the radical, solving the resulting equation, and checking for extraneous roots.
Tugas 5.6 kalkulus aplikasi integral tentu (luas bidang datar)Nurkhalifah Anwar
油
The document contains calculations to find the total area ((R)) of shaded regions bounded by curves over given intervals using integral calculus. Several examples are worked through step-by-step showing the setup and evaluation of definite integrals to obtain the shaded area values. The areas found include 8/3, 2, 42, /2, 128/15, 1/12, and 4/3 units.
This material is for PGPSE / CSE students of AFTERSCHOOOL. PGPSE / CSE are free online programme - open for all - free for all - to promote entrepreneurship and social entrepreneurship PGPSE is for those who want to transform the world. It is different from MBA, BBA, CFA, CA,CS,ICWA and other traditional programmes. It is based on self certification and based on self learning and guidance by mentors. It is for those who want to be entrepreneurs and social changers. Let us work together. Our basic idea is that KNOWLEDGE IS FREE & AND SHARE IT WITH THE WORLD
This document provides examples of factorizing algebraic expressions by finding the highest common factor (HCF) of the terms. It shows expressions being factorized, such as 2a+6 being written as 2(a+3), and 8m+12 being written as 4(2m+3). The document explains that algebraic expressions can sometimes be written as the HCF multiplied by grouped terms in parentheses. It provides steps for finding the factors of each term and the HCF to factorize expressions like 9jk+4k as k(9j+4).
Worried about due completion of math homework? You can count on us. In Homework1.com we have excellent infrastructure to serve you even at the most critical hours of assignment submission! Try our service today and get excellent score in math exam.
1) Completing the square allows quadratic expressions to be written in the form x + a^2 + b. This is useful for solving quadratic equations and finding the turning point of parabolas.
2) To solve the equation x^2 + 4x - 7 = 0 by completing the square:
a) Write the expression in the form x + a^2 + b as x + 2^2 - 11
b) Set this equal to 0 and solve for x, obtaining the solutions x = -2 賊 11.
3) Completing the square and writing quadratic expressions in the form x + a^2 + b allows them to be more easily solved and for properties like the
Strategic intervention materials on mathematics 2.0Brian Mary
油
This document provides teaching materials on solving quadratic equations by factoring for a mathematics class. It includes an overview of quadratic equations and their standard form. It then outlines least mastered skills and activities to practice identifying quadratic equations, rewriting them in standard form, factoring trinomials, and determining roots. Example problems and solutions are provided to demonstrate factoring trinomials and using factoring to solve quadratic equations. A practice problem asks students to solve a word problem involving a quadratic equation. Key terms and concepts are bolded. References for further reading are listed at the end.
This document summarizes key concepts about quadratic equations, including:
- Quadratic equations can be solved by factoring, completing the square, or using the quadratic formula.
- Completing the square involves manipulating the equation into a perfect square trinomial form.
- The quadratic formula provides the solutions to any quadratic equation in standard form.
- Cubic equations that are the sum or difference of cubes can be factored and solved.
- Literal quadratic equations can be solved for a specified variable using techniques like the square root property or quadratic formula.
- The discriminant determines whether the solutions to a quadratic equation are rational, irrational, or complex numbers.
The document discusses solving polynomial equations. It begins by explaining quadratic equations, including how to solve them by factoring or using the quadratic formula. It then introduces polynomial equations of higher degree and methods for determining their real or complex roots, including the fundamental theorem of algebra. Examples are provided to illustrate solving quadratic and polynomial equations using these various methods.
The document discusses solving quadratic equations in one variable of the form ax^2 + bx + c = 0. It provides examples of quadratic equations and shows how to rewrite them in standard form. It then covers methods for solving quadratic equations, including extracting square roots, factoring, completing the square, and using the quadratic formula. It also discusses the nature of the roots based on the discriminant and provides rules for determining the sum and product of the roots.
Here are the steps to solve this equation:
1) The equation is: x + 3 / 6 = 1
2) Multiply both sides by 6: x + 3 = 6
3) Subtract 3 from both sides: x = 3
Therefore, the solution is x = 3.
(d) The equation is: 3(y 7) = 14 y/2
3y 21 = 14 y/2
3y 21 + y/2 = 14
(12y 21)/6 = 14
12y 21 = 84
12y = 105
y = 9
The solution is y = 9.
(Check: if y =
This document discusses solving systems of equations by elimination. It provides examples of eliminating a variable by adding or subtracting equations. The key steps are: 1) write the equations in standard form; 2) add or subtract the equations to eliminate one variable; 3) substitute the eliminated variable back into one equation to solve for the other variable. Checking the solution in both original equations verifies the correct solution was found.
This document provides solutions to exercises from NCERT Class 9 Maths Chapter 4 on linear equations in two variables. It includes:
1) Solving linear equations representing word problems and expressing equations in the form ax + by + c = 0.
2) Finding solutions that satisfy given linear equations and determining the value of k if a given point is a solution.
3) Drawing graphs of various linear equations by plotting points that satisfy each equation.
4) Giving two equations of lines passing through a point and noting there are infinitely many such lines.
This document contains an exercise set with 46 problems involving real numbers, intervals, and inequalities. The problems cover topics such as determining whether numbers are rational or irrational, solving equations, graphing inequalities on number lines, factoring polynomials, and solving compound inequalities.
1. The document provides 10 problems to calculate the derivative (dy/dx) of various functions involving radicals, trigonometric functions, and composition of functions.
2. The solutions show setting the radical or function within a radical equal to a variable u, then calculating du/dx and applying the chain rule and power rule to determine dy/dx.
3. The answers are provided in fractional form with radicals, involving variables and parameters from the original functions.
This document provides information about an online math homework help service called Homework1. It includes:
- Contact information for Homework1, including their address, phone number, email, and social media links.
- An overview of the services provided, which include math homework help, writing assistance, and teaching students the solutions to help them learn.
- Several examples and solutions to common math problems to illustrate the type of homework help offered.
This document discusses how to solve radical equations by isolating the radical expression, removing the radical sign by raising both sides to the appropriate power, and solving the resulting equation. It provides examples of solving various radical equations step-by-step and checking solutions. Key steps include isolating the radical term, removing the radical, solving the resulting equation, and checking for extraneous roots.
Tugas 5.6 kalkulus aplikasi integral tentu (luas bidang datar)Nurkhalifah Anwar
油
The document contains calculations to find the total area ((R)) of shaded regions bounded by curves over given intervals using integral calculus. Several examples are worked through step-by-step showing the setup and evaluation of definite integrals to obtain the shaded area values. The areas found include 8/3, 2, 42, /2, 128/15, 1/12, and 4/3 units.
This material is for PGPSE / CSE students of AFTERSCHOOOL. PGPSE / CSE are free online programme - open for all - free for all - to promote entrepreneurship and social entrepreneurship PGPSE is for those who want to transform the world. It is different from MBA, BBA, CFA, CA,CS,ICWA and other traditional programmes. It is based on self certification and based on self learning and guidance by mentors. It is for those who want to be entrepreneurs and social changers. Let us work together. Our basic idea is that KNOWLEDGE IS FREE & AND SHARE IT WITH THE WORLD
This document provides examples of factorizing algebraic expressions by finding the highest common factor (HCF) of the terms. It shows expressions being factorized, such as 2a+6 being written as 2(a+3), and 8m+12 being written as 4(2m+3). The document explains that algebraic expressions can sometimes be written as the HCF multiplied by grouped terms in parentheses. It provides steps for finding the factors of each term and the HCF to factorize expressions like 9jk+4k as k(9j+4).
Worried about due completion of math homework? You can count on us. In Homework1.com we have excellent infrastructure to serve you even at the most critical hours of assignment submission! Try our service today and get excellent score in math exam.
1) Completing the square allows quadratic expressions to be written in the form x + a^2 + b. This is useful for solving quadratic equations and finding the turning point of parabolas.
2) To solve the equation x^2 + 4x - 7 = 0 by completing the square:
a) Write the expression in the form x + a^2 + b as x + 2^2 - 11
b) Set this equal to 0 and solve for x, obtaining the solutions x = -2 賊 11.
3) Completing the square and writing quadratic expressions in the form x + a^2 + b allows them to be more easily solved and for properties like the
Strategic intervention materials on mathematics 2.0Brian Mary
油
This document provides teaching materials on solving quadratic equations by factoring for a mathematics class. It includes an overview of quadratic equations and their standard form. It then outlines least mastered skills and activities to practice identifying quadratic equations, rewriting them in standard form, factoring trinomials, and determining roots. Example problems and solutions are provided to demonstrate factoring trinomials and using factoring to solve quadratic equations. A practice problem asks students to solve a word problem involving a quadratic equation. Key terms and concepts are bolded. References for further reading are listed at the end.
This document summarizes key concepts about quadratic equations, including:
- Quadratic equations can be solved by factoring, completing the square, or using the quadratic formula.
- Completing the square involves manipulating the equation into a perfect square trinomial form.
- The quadratic formula provides the solutions to any quadratic equation in standard form.
- Cubic equations that are the sum or difference of cubes can be factored and solved.
- Literal quadratic equations can be solved for a specified variable using techniques like the square root property or quadratic formula.
- The discriminant determines whether the solutions to a quadratic equation are rational, irrational, or complex numbers.
The document discusses solving polynomial equations. It begins by explaining quadratic equations, including how to solve them by factoring or using the quadratic formula. It then introduces polynomial equations of higher degree and methods for determining their real or complex roots, including the fundamental theorem of algebra. Examples are provided to illustrate solving quadratic and polynomial equations using these various methods.
The document discusses solving quadratic equations in one variable of the form ax^2 + bx + c = 0. It provides examples of quadratic equations and shows how to rewrite them in standard form. It then covers methods for solving quadratic equations, including extracting square roots, factoring, completing the square, and using the quadratic formula. It also discusses the nature of the roots based on the discriminant and provides rules for determining the sum and product of the roots.
Here are the steps to solve this equation:
1) The equation is: x + 3 / 6 = 1
2) Multiply both sides by 6: x + 3 = 6
3) Subtract 3 from both sides: x = 3
Therefore, the solution is x = 3.
(d) The equation is: 3(y 7) = 14 y/2
3y 21 = 14 y/2
3y 21 + y/2 = 14
(12y 21)/6 = 14
12y 21 = 84
12y = 105
y = 9
The solution is y = 9.
(Check: if y =
College algebra real mathematics real people 7th edition larson solutions manualJohnstonTBL
油
This document contains information about the College Algebra Real Mathematics Real People 7th Edition Larson textbook including:
- A link to download the solutions manual and test bank for the textbook
- An overview of the content covered in Chapter 2 on solving equations and inequalities, including linear equations, identities, conditionals, and more.
- 51 example problems from Chapter 2 with step-by-step solutions.
This document provides an overview of quadratic equations and inequalities. It defines quadratic equations as equations of the form ax2 + bx + c = 0, where a, b, and c are real number constants and a 0. Examples of quadratic equations are provided. Methods for solving quadratic equations are discussed, including factoring, completing the square, and the quadratic formula. Properties of inequalities are outlined. The chapter also covers solving polynomial and rational inequalities, as well as equations and inequalities involving absolute value. Practice problems are included at the end.
This document discusses quadratic equations and their properties. It defines quadratic equations as equations of the form y=ax^2 +bx + c, where the highest power is 2. It explains that quadratic equations can be solved using the quadratic formula, x = -b 賊 (b^2 - 4ac) / 2a. The number of solutions depends on the discriminant, b^2 - 4ac. If it is greater than 0, there are two solutions, if equal to 0 there is one solution, and if less than 0 there are no solutions. Examples are provided to demonstrate solving quadratic equations.
This document discusses methods for solving quadratic and cubic equations. It begins by introducing quadratic equations in standard form and methods for solving them, including factoring, completing the square, and using the quadratic formula. It then discusses properties related to the square root and applies them to solving quadratic equations. The document concludes by introducing cubic equations that are the sum or difference of cubes, and provides an example of solving one using factoring.
Produccion escrita expresiones algebraicasJuinAndresDiaz
油
The document discusses various algebraic operations including:
1) Summing algebraic expressions by adding like terms.
2) Subtracting algebraic expressions by adding the opposite of like terms.
3) Multiplying algebraic expressions by multiplying each term in one factor by each term in the other factor.
4) Dividing algebraic expressions using long division.
The document provides steps and examples for solving various types of word problems in algebra, including number, mixture, rate/time/distance, work, coin, and geometric problems. It also covers solving quadratic equations using methods like the square root property, completing the square, quadratic formula, factoring, and using the discriminant. Finally, it discusses linear inequalities, including properties related to addition, multiplication, division, and subtraction of inequalities.
The document contains examples and explanations of solving systems of equations by substitution. In Example 1, a system with two equations and two variables is solved to find the solution (2,4). In Example 2, a real-world word problem is modeled with a system of three equations with three variables to represent the number of different types of tickets printed for a play. The system is solved to find the numbers of adult (A=500), student (S=1000), and children's (C=250) tickets printed.
Expresiones Algebraicas, Factorizaci坦n y Radicaci坦nkarladiazperaza
油
The document discusses algebraic expressions including:
- Adding and subtracting algebraic expressions by combining like terms
- Finding the numerical value of an algebraic expression by substituting a number for the variable
- Multiplying algebraic expressions by using the rules of exponents
- Dividing algebraic expressions by using the inverse rules of exponents
- Factorizing algebraic expressions using factoring techniques like common factors, difference of squares, and perfect square trinomials.
1. This document discusses solving quadratic equations by factorizing, using the quadratic formula, and finding the roots given information about the sum and product of the roots.
2. Methods covered include factorizing quadratic expressions, setting each factor equal to zero to find roots, using the quadratic formula, and deriving the quadratic equation when given the sum and product of the roots.
3. Examples are provided to demonstrate each method, such as factorizing (x-1)(x+2)=1 to find the roots x=1 and x=-2, or using the quadratic formula to solve equations like x2-3x-10=0.
Recognize features of systematic reviews and meta-analyses as a research design
Identify the elements of a well-defined review question
Understand and develop search strategies and able to turn research questions into search strategy
Perform a comprehensive search for relevant studies
Manage the results of systematic searches
Extract data and assess risk of bias of included studies
Understand and carry out quantitative analysis of extracted data
Apply the methodology and conduct reviews independently
How to Configure Outgoing and Incoming mail servers in Odoo 18Celine George
油
Odoo 18 features a powerful email management system designed to streamline business communications directly within the platform. By setting up Outgoing Mail Servers, users can effortlessly send emails. Similarly, configuring Incoming Mail Servers enables Odoo to process incoming emails and generate records such as leads or helpdesk tickets.
Stages of combustion, Ignition lag, Flame propagation, Factors affecting flame
speed, Abnormal combustion, Influence of engine design and operating
variables on detonation, Fuel rating, Octane number, Fuel additives, HUCR,
Requirements of combustion chambers of S.I. Engines and its types.
Tollywood Quiz- 21st March 2025, Quiz Club NITWQuiz Club NITW
油
The most anticipated Tollywood Quiz, organised by the Quiz Club NITW, was held on March 21, 2025. The quiz set will take you on a nostalgic journey through iconic movies and their unforgettable songs and dialogues.
Design approaches and ethical challenges in Artificial Intelligence tools for...Yannis
油
The recent technology of Generative Artificial Intelligence (GenAI) has undeniable advantages, especially with regard to improving the efficiency of all stakeholders in the education process.
At the same time, almost all responsible international organisations and experts in the field of education and educational technology point out a multitude of general ethical problems that need to be addressed. Many of these problems have already arisen in previous models of artificial intelligence or even in systems based on learning data, and several are appearing for the first time.
In this short contribution, we will briefly review some dimensions of ethical problems, both (a) the general ones related to trust, transparency, privacy, personal data security, accountability, environmental responsibility, bias, power imbalance, etc., and (b) the more directly related to teaching, learning, and education, such as students' critical thinking, the social role of education, the development of teachers' professional competences, etc.
In addition, the categorizations of possible service allocation to humans and AI tools, the human-centered approach to designing AI tools and learning data, as well as the more general design of ethics-aware applications and activities will be briefly presented. Finally, some short illustrative examples will be presented to set the basis for the debate in relation to ethical and other dilemmas.
Unit1 Inroduction to Internal Combustion EnginesNileshKumbhar21
油
Introduction of I. C. Engines, Types of engine, working of engine, Nomenclature of engine, Otto cycle, Diesel cycle Fuel air cycles Characteristics of fuel - air mixtures Actual cycles, Valve timing diagram for high and low speed engine, Port timing diagram
Celine Caira presents at Women girls and AI Paving the way to a balanced digi...EduSkills OECD
油
Celine Caira, Economist & Policy Analyst, AI Unit of the OECD Division of Science, Technology and Innovation (STI), OECD presents at the OECD Webinar 'Women, girls and AI: Paving the way to a balanced digital future' on 28 March 2025. PPT by Lucia Russo, B辿n辿dicte Rispal and
Celine Caira OECD
Measles OutbreakSouthwestern US This briefing reviews the current situation surrounding the measles outbreaks in Texas, New Mexico, Oklahoma, and Kansas.
How to Install Odoo 18 with Pycharm - Odoo 18 際際滷sCeline George
油
In this slide well discuss the installation of odoo 18 with pycharm. Odoo 18 is a powerful business management software known for its enhanced features and ability to streamline operations. Built with Python 3.10+ for the backend and PostgreSQL as its database, it provides a reliable and efficient system.
S. Y. G. N. M. CHILD HEALTH NURSING Leukemia in Children.pptxsachin7989
油
Leukemia is a type of cancer that affects the blood and bone marrow. It occurs when abnormal white blood cells accumulate in the bone marrow and interfere with the production of normal blood cells.
Types of Leukemia
1. Acute Lymphoblastic Leukemia (ALL): A type of leukemia that affects the lymphoid cells.
2. Acute Myeloid Leukemia (AML): A type of leukemia that affects the myeloid cells.
3. Chronic Lymphocytic Leukemia (CLL): A type of leukemia that affects the lymphoid cells and progresses slowly.
4. Chronic Myeloid Leukemia (CML): A type of leukemia that affects the myeloid cells and progresses slowly.
Symptoms of Leukemia
1. Fatigue
2. Weight loss
3. Pale skin
4. Bruising or bleeding easily
5. Bone or joint pain
6. Swollen lymph nodes
7. Loss of appetite
Treatment of Leukemia
1. Chemotherapy
2. Radiation therapy
3. Bone marrow transplant
4. Targeted therapy
Thalassemia
Thalassemia is a genetic disorder that affects the production of hemoglobin, a protein in red blood cells that carries oxygen to the body's tissues.
Types of Thalassemia
1. Alpha-Thalassemia: A type of thalassemia that affects the production of alpha-globin chains.
2. Beta-Thalassemia: A type of thalassemia that affects the production of beta-globin chains.
Symptoms of Thalassemia
1. Anemia
2. Fatigue
3. Pale skin
4. Shortness of breath
5. Enlarged spleen
6. Bone deformities
Treatment of Thalassemia
1. Blood transfusions
2. Iron chelation therapy
3. Bone marrow transplant
4. Gene therapy
1. Homework1
Copyright 息 2014-2016 Homework1.com, All rights reserved
Math Homework Help | Math Homework Help Service
Contact Us
Homework1
3422 SW 15 Street
Suite #8924
Deerfield Beach, FL, US 33442
Tel: +1-626-472-1732
Web: https://homework1.com/
Email: info@homework1.com
Facebook: https://www.facebook.com/homework1com
Linkedin: https://www.linkedin.com/in/homework1
Twitter: https://twitter.com/homework1_com
Google Plus: https://plus.google.com/+Homework1/
Pinterest: https://www.pinterest.com/homeworkone/
2. Homework1
Copyright 息 2014-2016 Homework1.com, All rights reserved
About Us:
At Homework1.com we offer authentic and 100% accurate
online homework help and study assistance to students from
USA, UK, Australia, and Canada. However, we dont offer
students only academic assignment help service to complete
their study project; rather we offer our best effort to teach our
student-clients about the assignment we have solved. Our
tutors are not only subject matter experts, they are avid
student-mentors and are ready to walk extra miles to make
them understand the fundamentals of the assignment done,
and help them to learn the solution by heart.
We are always ready to hear from you as you will find us always a few clicks away in
247! We are reachable by a phone calls you can send us an email or simply but
accessing our site you can call us for a live chat! We are ready to help you at the most
critical hour of your assignment submission and will take care of your task with best
sincerity and prompt turnaround time! Help with math homework service proves to
be beneficial at exam times and assignments having short deadlines.
Sample of Math Homework Help Illustrations and Solutions:
Example: If the root of the equation 22
- 10x + = 0 = is 2, then the value of is :
(a) -3 (b) -6, (c) 9 (d) 12
Solution. Since is a root of the equation
2(2)2
- 10(2) + =0 = = 20 8 = 12 (d) holds.
Example: Which of the following is a root of the equation 22
- 5x 3 = 0 ?
(a) x = 3 (b) x = 4 (c) x = 1 (d) x = -4
Solution. (a) holds. ( 2(3)2
-5(3) 3 = 0 = 18 15 3 = 0)
Example: If no root of 2
-kx + 1 = 0 is real, then
(a) -3 < k < 3 (b) 2 <k <2 (c) k > 2 (d) k < -2
(b) Solution. Since the roots of 2
kx + 4 = 0 are non-real.
Disc. ()2
- 4 < 0 = 2
- 4 < 0 = 2
< 4 = |k| <2 (b) holds
Example: If 2
+ 4x + k = 0 has real roots, then
(a) -3 < k < 3 (b) -2 <k <2 (c) k > 2 (d) k < -2
(C.B.S.E. 2012)
Solution. Since the roots of 2
- kx + 4 = 0 are non-real.
Disc. ()2
-4 < 0 = 2
< 4 = |k| < 2 = -2 < k < 2 (b) holds.
Example: If 2
+ 4x + k = 0 has real roots, then
3. Homework1
Copyright 息 2014-2016 Homework1.com, All rights reserved
(a) k 4 (b) k 4 (c) k 0 (d) k 0
Solution. Since 2
+ 4x + k = has real roots.
Disc. (4)2
4k 0 = 4k 16 = k 4 (b) holds.
Example: Value of k for which the quadratic equation 22
- kx + k = 0 has equal roots is
(a) 0 only (b)4 (c) 8 only (d) 0.8.
(N.C.E.R.T. Exemplar Problem)
Solution. For equal roots, Disc. = ()2
-4(2) (k) = 0 = k (k 8) = 0 = k = 0.8
(d) holds.
Example: The value of k for which 32
+ 2x + k = 0 has real roots is :
(a) k >
1
3
(b) k
1
3
(c) k
1
3
(d) k <
1
3
Solution. For real roots, Disc. = (2)2
4(3) (k) 0
= 4 12k 0 = 4 12 k = 12 k 4 = k
4
12
=
1
13
= (b) holds.
Example: If the quadratic equation 2
+ 2x + m = 0 has two equal roots, then the
values of m are
(a) 賊 1 (b) 0,-2 (c) 0,1 (d) -1,0
4. Homework1
Copyright 息 2014-2016 Homework1.com, All rights reserved
Solution. Disc. = 2
- 4 (6) (2) = 1 (given) = 2
= 48 + 1 = 49 =b = 賊 7 (c) holds.
Solved examples 6
Example: Find the sum and product of roots of (-2)2
+ 5x + 4 = 0.
Solution. Sum of roots =
=
5
2
=
5
2
; product of roots =
=
4
2
=-2
Example: If and are roots of 92
- 24x + 8 = 0, then find + and 腫.
Solution. + =
=
(24)
9
=
8
3
; 腫 =
=
8
9
Example: Form a quadratic equation roots -
1
3
and
5
2
.
Solution. S = Sum of roots =
1
3
+
5
2
=
2+15
6
; P = Product of roots =
1
3
5
2
= -
5
6
equation is 2
Sx + p = 0 = 2
-
13
6
x -
5
6
= 0 = 62
-13x 5 = 0
Example: If one root of quadratic equation with rational co-efficients is 2 + 3 , then
give other root.
5. Homework1
Copyright 息 2014-2016 Homework1.com, All rights reserved
Solution. Note that in a quadratic equation with rational co-efficients, surd roots occur
in conjugate pairs. = 2 - 3 is the other root.
1. The sum of the squares of two positive integers is 208. If the square of the larger
number is 18 times the smaller, find the numbers.
Solution. Let the smaller number = x ; Square of larger number = 18x
Given, 2
+ 18x = 208 = 2
+ 18x 208 = 0
= (x 8) (x + 26) = 0 =x = 8, -26 = x = 8
Square of larger number = 18x = (18) (8) = 144
= Larger number = 144 = 12 and smaller number = x = 8
2. If -5 is root of quadratic equation 22
+ 2px 15 = 0 and the quadratic equation
(2
+ x) k = 0 has equal roots, find the value of k.
Solution. 5 is root of 22
+ 2px 15 = 0 = 2 (5)2
+ 2 (-5) 15 = 0 = =
7
2
(2
+ x) + k = 0 =
7
2
(2
+ x) + k = 0 = 72
+ 7x + 2k = 0
It has equal roots = D = 0 = (7)2
-4(7) (2k) = 0 = k =
7
8
3. If, p, q, r and s are real numbers such that pr = 2(q + s), then show that at least
one of the equations 2
+ px + q = 0 has real roots.
4. If the roots of the equation 2
+ 2cx + ab = 0 are real and unequal, prove that
the equation 2
-2 (a + b) x + 2
+ 2
+ 22
= 0 has no real roots.
5. A person on tour has $360 for his expenses. If the extends his tour for 4 days, he
has to cut down his daily expenses by $3. Find the original duration of the tour.
6. Homework1
Copyright 息 2014-2016 Homework1.com, All rights reserved
Solution. Let the original duration of the tour be x days.
total expenditure on tour = $360 expenditure per day = $
360
Duration of the extended tour = (x + 4) days
expenditure per day according to new schedule = $
360
+4
Since the daily expenses are cut down by $3.
360
-
360
+4
= 3 =
360 +4 360
(+4)
= 3
=
360 +1440360
(+4)
= 3 =
1440
2+ 4
= 3 = 2
+ 4x = 480 = 2
+ 4x 480 = 0
= 2
+ 24x 20x 480 = 0 = x (x + 24) -20 (x + 24) = 0
= (x 20) (x + 24) = 0 = x 20 = 0 or, x + 24 = 0 =x = 20 or, x = -24
But, the number of days cannot be negative. So, x = 20
Hence, the original duration of the tour was of 20 days.
6. (a) A shopkeeper buys a number of books for $80. If he had bought 4 more
books for the same amount, each book would have cost $12 less. How many
books did he buy?
7. Homework1
Copyright 息 2014-2016 Homework1.com, All rights reserved
(b) A shopkeeper buys a number of books for $1200. If he had bought 10 more books
for the same amount, each book would have cost him $20 less. How many books did he
buy ?
Solution. (a) Let number of books bought be x. Then, Cost of x books = $80
= Cost of one book = $
80
If the number of books bought is x + 4, then Cost of one book = $
80
+4
It is given that the cost of one book is reduced by one rupee
80
-
80
+4
= 1 = 80
1
1
+4
= 1 = 80
+4
+4
= 1 =
320
2+ 4
= 1
= 2
+ 4x = 320 = 0 = 2
+ 20x 16x 320 = 0 = x (x + 20) 16(x + 20) = 0
= (x + 20) (x 16) = 0 =x = -20 or, x = 16 = x = 16 [ x cannot be negative]
Hence, the number of books is 16.
(b) Similar to Part (a) Ans. 20.
7. If the price of a book is reduced by $5, a person can buy 5 more books for $300. Find
the original list price of the book.
Solution. Let the original list price of the book be $ x number of books bought for
$300 =
300
Reduced list price of the book = $(x -5) number of books bought for $300 =
300
モ5
By the given condition,
300
モ5
-
300
= 5 =
300 モ300 +1500
(モ5)
= 5 =
1500
25
= 5
= 2
- 5x = 300 = 2
- 5x 300 = 0 = 2
- 20x + 15x 300 = 0 = (x 20) (x + 15) =
0
= x 20 = 0 or, x + 15 = 0 = x = 20, x = -15 =x = 20 [ x = -15 is not possible]
Hence, the list price of the books = $20
8. A factory kept increasing is doubled in the last two year. Find the percentage if it is
known that the output is doubled in the last two years.
Solution. Let p be the initial production (2 years ago), and let the increase in product
every year be x%. Then,
Product at the end of first year = P +
100
= p 1 +
100
Product at the end of the second year
= P 1 +
100
+
100
1 +
100
= P 1 +
100
1 +
100
= P 1 +
100
2
8. Homework1
Copyright 息 2014-2016 Homework1.com, All rights reserved
Since product is doubled in last two years P 1 +
100
2
= 2P = 1 +
100
2
= 2
= (100 + )2
= 2 1002
= 2
+ 200x 10000 = 0
= x =
200 賊 (200)2+ 40000
2
= - 100 賊 100 2 = 100 (-1 + 2)
= x = 100 (-1 + 2) [ x cannot be negative]
9. The difference of the ages of Sohrab and his father is 30 years. If the difference of the
squares of their ages is 1560, find their ages.
Solution. Let y, x be the ages of Soharb and his father respectively
x y = 30 (1)
and 2
- 2
= 1560 (2)
Divide (2) by (1), we get
2 2
モ
=
1560
30
= x + y = 52 (3)
(1) + (2) gives 2x = 82 =x = 41. (3) (1) gives 2y = 22 = y =
11
Hence required ages are 41 years, 11 years.