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Date Created: 11/09/15
Running Head: PTYHAGOREAN QUADRATIC 1 Pythagorean Quadratic name MAT 221: Introduction to Algebra Linda Seeger March 30, 2014 PYTHAGOREAN QUADRATIC 2 Pythagorean Quadratic In our fast pasted world people utilize GPS more than traditional road maps. One reason is that GPS uses trilateration to pinpoint someone's location and that of the location that they want to get to. The same method could be used to get the directions on a road map. Trilateration uses a formula similar to the Pythagorean Theorem formula. “The Pythagorean theorem says that in any right triangle the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse” (Dugopolski, 2012, p. 366). The Pythagorean Theorem is used most commonly in building and locating, it is the method that is being used to narrow down the possible location of Malaysian Airlines flight 370. Since the Pythagorean Theorem can be used to find a location on a treasure map. We are tasked with finding how many paces does x equal. 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? (Dugopolski , 2012, p. 371). From the information we are given we understand that Vanessa’s paces form the legs of a right triangle when sketched out on paper and 2x + 6 are the length of the hypotenuse. Now we use the Pythagorean Theorem to find out how many paces x is. The Pythagorean Theorem states that in every right triangle with legs of length a and b and hypotenuse c, these lengths have the formula 2 2 2 of a + b = c . PYTHAGOREAN QUADRATIC 3 Pythagorean Quadratic Incorporating the given information a = x, and b = 2x + 4, so that c = 2x + 6 to get: x + (2x + 4) = (2x + 6) 2 x + 4x + 16x + 16 = 4x + 24x + 36 Next the binomials are squared and subtract 4x 2 from both sides. x + 16x + 16 = 24x + 36. Then 16x is subtracted from both sides x +16 = 8x + 36. We subtract 36 from both sides 2 x 20= 8x. Last 8x is subtracted from both sides leaving 2 x – 8x – 20 =0. A quadratic equation that can be solved by factoring and using the zero factor. (x – ) (x + ) = 0 The coefficient of x is 1 we start with x in a pair of parenthesis with one being negative and the other positive because 20 is negative. (x – 10)(x + 2) = 0 Next I looked for two numbers that when multiplied will yield 20 and add up to 8 which is 10 + 2 the incorporate them in the problem and use the zero factor to solve each binomial. x– 10 = 0 or x + 2 = 0 This creates a compound equation where we solve for x x = 10 or x = 2 The possible solutions to the equation PYTHAGOREAN QUADRATIC 4 One of the solutions does not work in this scenario. Since paces and distances cannot be negative in a geometric figure, 2 is the solution that does not work. We are left with x = 10 as the number of paces so now we know that the treasure lies 10 paces north and 2x + 4 = 2(10) + 4 = 24 paces to the east or 2x + 6 = 2(10) + 6 = 26 paces straight from Castle Rock. Pythagorean Quadratic Utilization of the Pythagorean Theorem is important in locating a given point. The same triangulation used in the evaluating a Pythagorean Theorem is used in GPS systems to give you directions to an address. By solving for x, the number of paces was found. Using the Pythagorean Theorem formula along with the given information x was solved for and the map would be able to guide Vanessa and Ahmed to the treasure. PYTHAGOREAN QUADRATIC 5 Reference Dugopolski, M. (2012). Elementary and intermediate algebra (4th ed.). New York, NY: McGrawHill Publishing.
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