Pythagorean QuadraticUsing the Problem: Ahmed has half of a treasure map,which indicates that the treasure is buried in the desert 2x + 6 paces from Castle Rock. Vanessa has the other half of the map. Her half indicates that to find the treasure, one must get to Castle Rock, walk x paces to the north, and then walk 2x + 4 paces to the east. If they share their information, then they can find x and save a lot of digging. What is x?Along with factoring, this problem also requires the use of the Pythagorean Theorem to solve it.Read the following instructions in order to complete his assignment and review the example of how to complete the math required for this assignment, and review the example of how to complete the math required for this assignment: Read problem 98 on page 371 of Elementary and Intermediate Algebra. Along with factoring, this problem also requires the use of the Pythagorean Theorem to solve it.On your scratch paper, draw a diagram which includes the location of Castle Rock and the dimensions of a triangle with the given number of paces. Seeing the relationship of the sides and where the right angle is will help you know how to apply the Theorem. You may include the diagram in your paper but are not required to do so.Write the equation first by inserting the given binomials into the Pythagorean Theorem. Once you have your equation written, solve it using the methods from section 5.6 of Elementary and Intermediate Algebra. Show all steps and explain what you are doing as you go.Write a two- to three-page paper that is formatted in APA style and according to the Math Writing Guide. Format your math work as shown in the Instructor Guidance and be concise in your reasoning. In the body of your essay, do the following:Demonstrate your solution to the above problem, making sure to include all mathematical work and an explanation for each step.Explain which solution to the equation is the best solution and why it is the best solution. How many steps is the map from Castle Rock?Incorporate the following four math vocabulary words into your writing. Use bold font to emphasize these words in your writing (Do not write definitions for the words; use them appropriately in sentences describing your math work):QuadraticZero factorPythagorean TheoremCompound equationRunning 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.1Pythagorean Quadratic (full title; centered horizontally & vertically) First Name Last Name MAT 221 Dr. xxxxxxxxxxx xxxxxxxxx DatePYTHAGOREAN QUADRATIC2Pythagorean 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 + 9x2x + 6PYTHAGOREAN QUADRATIC 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.3?2 + 4?2 + 24x + 36 = 4?2 + 36x + 81 The binomials squared. Notice there is a 4?2 -4?2 -4?2 on both sides of the equation that can be subtracted out first. x2 + 24x + 36 = 36x + 81 -36x -36x Subtract 81 from both sides of equation. Subtract 36x from both sides of equation.x2 ? 12x + 36 = 81 -81 -81?2 ? 12x ? 45 = 0Remaining is a quadratic equation to solve by factoring and using the zero factor.(x ? ) (x + ) = 0Since the coefficient of x2 is 1, start with a pair of parentheseswith 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 x ? 15 = 0 or x + 3 = 0 +15 +15 -3 -3 Use the zero factor property to solve each binomial, creating a compound equation. Solve for x using equation rules. These are the possible solutions to our equation.x = 15 or x = -3PYTHAGOREAN 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.4[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.]PYTHAGOREAN QUADRATIC Reference Dugopolski, M. (2012). Elementary and intermediate algebra (4th ed.). New York, NY: McGraw-Hill Publishing.5Use 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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