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This module is from Elementary Algebra</link>by Denny Burzynski and Wade Ellis, Jr. Methods of solving quadratic equations as well as the logic underlying each method are discussed. Factoring, extraction of roots, completing the square, and the quadratic formula are carefully developed. The zero-factor property of real numbers is reintroduced. The chapter also includes graphs of quadratic equations based on the standard parabola, y = x^2, and applied problems from the areas of manufacturing, population, physics, geometry, mathematics (numbers and volumes), and astronomy, which are solved using the five-step method.Objectives of this module: become more proficient at using the five-step method for solving applied problems.

Overview

  • The Five-Step Method
  • Examples

The five-step method

We are now in a position to study some applications of quadratic equations. Quadratic equations can arise from a variety of physical (applied) and mathematical (logical) problems.

We will, again, apply the five-step method for solving word problems.

Five-step method of solving word problems

  • Step 1:   Let x (or some other letter) represent the unknown quantity.
  • Step 2:   Translate the verbal expression to mathematical symbols and form an equation.
  • Step 3:   Solve this equation.
  • Step 4:   Check the solution by substituting the result into the equation found in step 2.
  • Step 5:   Write a conclusion.

Remember, step 1 is very important.

ALWAYS START BY INTRODUCING A VARIABLE.

Once the quadratic equation is developed (step 2), try to solve it by factoring. If factoring doesn’t work, use the quadratic formula. A calculator may help to make some of the calculations a little less tedious.

Sample set a

A producer of personal computer mouse covers determines that the number N of covers sold is related to the price x of a cover by N = 35 x x 2 . At what price should the producer price a mouse cover in order to sell 216 of them?

Step  1 : Let x = the price of a mouse cover . Step  2 : Since N is to be 216, the equation is 216 = 35 x x 2 Step  3 : 216 = 35 x x 2 Rewrite in standard form . x 2 35 x + 216 = 0 Try factoring . ( x 8 ) ( x 27 ) = 0 x 8 = 0 o r x 27 = 0 x = 8 o r x = 27 Check these potential solutions . Step  4 : If x = 8 , If x = 27 , 35 · 8 8 2 = 216 Is this correct? 35 · 27 27 2 = 216 Is this correct? 280 64 = 216 Is this correct? 945 729 = 216 Is this correct? 216 = 216 Yes, this is correct . 216 = 216 Yes, this is correct . These solutions check . Step 5 : The computer mouse covers can be priced at either $8 or $27 in order to sell 216 of them .

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Practice set a

A manufacturer of cloth personal computer dust covers notices that the number N of covers sold is related to the price of covers by N = 30 x x 2 . At what price should the manufacturer price the covers in order to sell 216 of them?

Step 1:


Step 2:


Step 3:




Step 4:




Step 5:   In order to sell 216 covers, the manufacturer should price them at either or .

12 or 18

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It is estimated that t years from now the population of a particular city will be

P = t 2 24 t + 96 , 000.

How many years from now will the population be 95,865?

Step 1:


Step 2:


Step 3:




Step 4:




Step 5:

In 9 and 15 years, the population of the city will be 95,865.

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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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