MODULE 5

• DISCIPLINE • GOOD TASTE • EXCELLENCE
• Writer: Daren V. Perez
• Cover Illustrator: Joel J. Estudillo
• Department of Education National Capital Region SCHOOLS DIVISION OFFICE MARIKINA CITY
• MATHEMATICS Quarter 1: Module 5 Applications of Quadratic Equations and Rational Algebraic Equations
• Quadratic equations


  Equations that contain a variable raised to the second power
• Rational algebraic equations


  Equations that contain rational expressions
• In this module, you will learn how to: Solve problems involving quadratic equations and rational algebraic equations
• You can say that you have understood the lesson in this module if you can already: 1. Solve problems involving quadratic equations; and 2. Solve problems involving rational algebraic equations
• Solving problems involving quadratic equations

  1. Represent the unknown quantity or quantities using a variable or variables

  2. Use the given facts to illustrate the problem

  3. Solve the mathematical equation

  4. Check the results
• When analysing markets, a range of assumptions are made about the rationality of economic agents involved in the transactions
• The Wealth of Nations was written

  1776
• Rational


  (in classical economic theory) economic agents are able to consider the outcome of their choices and recognise the net benefits of each one
• Rational agents will select the choice which presents the highest benefits
• Producers act rationally by


  Selling goods/services in a way that maximises their profits
• Workers act rationally by


  Balancing welfare at work with consideration of both pay and benefits
• Governments act rationally by


  Placing the interests of the people they serve first in order to maximise their welfare
• Rationality in classical economic theory is a flawed assumption as people usually don't act rationally
• Marginal utility


  The additional utility (satisfaction) gained from the consumption of an additional product
• If you add up marginal utility for each unit you get total utility
• The difference of two numbers is 2 and their product is 224. Find the numbers.
• Solving the problem

  1. Represent the unknown quantities using variables

  2. Set up an equation using the given information

  3. Solve the equation

  4. Check the solution
• The area of a rectangle is 560 cm2. The length is 3 more than twice the width.
• The product of two consecutive odd integers is 255. What is the value of the greater odd integer?
• If x^2 is added to 7x, the sum is -10. Which could be the value of x?
• The area of the concrete pathway is 350 m2 and its perimeter is 90 m. What is the length of the pathway?
• Working together, Angelo and Jason can paint their house in 2 days. Working alone, Angelo can do the job in 3 days less than Jason's time in working alone.
• After completing a journey of 84 km, a cyclist noticed that he would take 5 hours less if he could travel at a speed which is 5 kph more, what was the speed of the cyclist in kph?
• The difference in the average speed of two trains is 16 kph. The slower train takes 2 hours longer to travel 170 kilometers than the faster train takes to travel 150 kilometers. Find the speed of the slower train.
• An 8-hour river cruise goes 12 km upstream and then back. The speed of the river current is 2 kph. What is the speed of the boat in still water?
• Two faucets can fill a tank in 1 hour and 20 minutes. The first faucet takes more than two hours longer to fill the same tank when functioning without the second tap. How long does it take to fill each one separately?
• LCD


  Least Common Denominator
• Reciprocal


  The inverse of a number
• Solving for the time it takes for a job to be completed

  1. Find the equation that represents the relationship between the time taken by each person and the total time

  2. Solve the equation to find the time taken by each person
• Uniform speed


  Speed that does not change
• Solving for the speed of a train


  1. Set up equations for the two different speeds

  2. Solve the quadratic equation to find the original speed
• Solving for the time it takes each person to complete a job

  1. Find the equation that represents the relationship between the time taken by each person and the total time

  2. Solve the quadratic equation to find the time taken by each person
• Rational equation


  An equation that can be written as a ratio of two polynomials
• Solving for the speed of a motorboat


  1. Set up the rational equation that represents the time taken for the downstream and upstream journey

  2. Solve the equation to find the speed of the stream
• The LCD of 2/(x-1) and 3/(x+1) is x^2 - 1