Running Head: PYTHAGOREAN QUADRATIC Running head should use a shortened version of the title if the title is long! All capital letters for the title and the words Running and Head should be capitalized as well. Pythagorean Quadratic (full title; centered horizontally & vertically) First Name Last Name MAT 221 Dr. xxxxxxxxxxx xxxxxxxxx Date 1 PYTHAGOREAN QUADRATIC 2 Pythagorean Quadratic Be sure to have a centered title on page 1 of your papers!! [The introductory paragraph must be written by each individual student and the content will vary depending on what the student decides to focus on in the general information of the topic. YOUR INTRODUCTION SHOULD CONNECT MATH CONCEPTS AND REAL-WORLD APPLICATIONS. DO NOT INCLUDE THE DIRECTIONS IN THE INTRO! The following paragraph is not an introduction to the paper but rather the beginning of the assignment.] Here is a treasure hunting problem very similar to the one in the textbook (Dugopolski, 2012). This problem involves using the Pythagorean Theorem to find distance between several points. Spanky has half of a treasure map, which indicates treasure is buried 2x + 9 paces from Leaning Rock. Buckwheat has the other half of the treasure map, which says that to find the treasure one must walk x paces to the north from Leaning Rock and then 2x + 6 paces east. Spanky and Buckwheat found that with both bits of information they can solve for x and save themselves a lot of digging. How many paces is x? Even though Spanky’s half of the map does not indicate in which direction the 2x + 9 paces should go, it can be assumed that his and Buckwheat’s paces should end up in the same place. When sketched on scratch paper, a right triangle is formed with 2x + 9 being the length of the hypotenuse, and x and 2x + 6 being the legs of the triangle. When a right triangle is involved, the Pythagorean Theorem helps solve for x. 2x + 9 x 2x + 6 PYTHAGOREAN QUADRATIC 3 The Pythagorean Theorem states that in every right triangle with legs of length a and b and hypotenuse c, these lengths have the relationship of a2 + b2 = c2. Let a = x, and b = 2x + 6, so that c = 2x + 9. Then, putting these measurements into the Theorem, the equation becomes: x2 + (2x + 6)2 = (2x + 9)2 The binomials into the Pythagorean Theorem. x2 + 4x2 + 24x + 36 = 4x2 + 36x + 81 The binomials squared. Notice there is a 4x2 -4x2 -4x2 on both sides of the equation that can be subtracted out first. x2 + 24x + 36 = 36x + 81 -36x Subtract 36x from both sides of equation. -36x x2 – 12x + 36 = 81 -81 Subtract 81 from both sides of equation. -81 x2 – 12x – 45 = 0 Remaining is a quadratic equation to solve by factoring and using the zero factor. Since the coefficient of x2 is 1, start with a pair of parentheses (x – ) (x + ) = 0 with an x in each. Since the 45 is negative, there will be one + and one – in the binomials. Two factors of -45 that add up to -12 are needed as possible values. -1, 45; -3, 15; -5, 9; 1, -45; 3, -15; 5, -9. Looks like 3, -15 will do it because 3 + (-15) = -12. (x – 15)(x + 3) = 0 Use the zero factor property to solve each binomial, x – 15 = 0 or x + 3 = 0 creating a compound equation. +15 Solve for x using equation rules. +15 -3 x = 15 or x = -3 -3 These are the possible solutions to our equation. PYTHAGOREAN QUADRATIC However, one of these solutions is what we call extraneous because it does not work with this scenario at all. You cannot have negative paces or negative distance in a measured geometric figure; so the -3 solution does not work, leaving us with x = 15 as the key number of paces. The treasure lies 15 paces north and 2x + 6 = 2(15) + 6 = 36 paces east of Leaning Rock, or 2x + 9 = 2(15) + 9 = 39 paces straight from the rock. [The conclusion paragraph must be written by each individual student and the content will vary depending on what the student decides to include in their summary. DO NOT INCLUDE PERSONAL NARRATIVE LIKE “I LEARNED…” OR “WE CAN DO….”. BE SURE TO SUMMARIZE WHY THE CONCEPT(S) ARE IMPORTANT AND HOW THEY CAN BE CONNECTED TO OTHER CONCEPTS.] 4 PYTHAGOREAN QUADRATIC 5 Reference Dugopolski, M. (2012). Elementary and intermediate algebra (4th ed.). New York, NY: McGraw-Hill Publishing. Use the word ‘Reference’ or ‘References’ as the title. THERE IS NO COLON OR UNDERLINE. Textbook should ALWAYS be included in every assignment! Be sure to use appropriate indentation (hanging), font (Arial or Times New Roman), and size (12).
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