Pythagorean Quadratic
MAT221 Introduction to Algebra Pythagorean Quadratic
Week five of this class has been a complete challenge for me, from start to finish. Trying to master everything that we have been taught over the five weeks has truly been a test. I know there are benefits to knowing these principals, however, it stresses me to think about having to use it in real life circumstances.
This problem involves using the Pythagorean Theorem to find distance between several points in our textbook (Dugopolski, 2012). 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?
We need to look at the equation so we can know how far Ahmed will have to walk, which is 2x+6 paces from Castle Rock. Even though Vanessa’s half of the map does not indicate in which direction the 2x + 4 paces should go, it can be assumed that her’s and Ahmed’s paces should end up in the same place. When sketched on scratch paper, a right triangle is formed with 2x + 6 being the length of the hypotenuse, and x and 2x + 4 being the legs of the triangle. When a right triangle is involved, the Pythagorean Theorem helps solve for x.

The Pyhagorean Theorem states that in every right triangle with legs of length a and b and hypotenuse c, these length have the relationship of a2 + b2 =c2. Let a = x, and b = 2x + 6, so that c = 2x + 4 Then, putting these measurements into the Theorem, the equation becomes: x2 + (2x + 6)2 = (2x + 4)2 The binomials into the Pythagorean Theorem. x2 + 4x2 + 16x + 16 = 4x2 + 24x + 36 The binomials squared. Notice there is a 4x2 - 4x2 - 4x2 on both sides of the equation that can be subtracted out first. x2 + 16x + 16 = 24x + 36 Subtract 24x from both sides of equation - 24x - 24x x2 – 8x + 16 = 36 Subtract 16 from both sides of equation - 16 - 16 x2 – 8x – 20 = 0 Remaining is a quadratic equation to solve by factoring and using the zero factor.
(x - ) (x + ) = 0 Since the coefficient of x2 is 1, start with a pair of parentheses with x in each. Since the 20 is negative, there will be one + and one – in the binomials. Two factors of – 20 that add up to – 8 are needed as possible values.
- 4, 5; - 10, 2 Looks like – 10, 2 will do because 2 + (-10) = - 8.
(x – 10)(x + 2) Use the zero factor property to solve each binomial, x – 10 = 0 or x + 2 = 0 creating a compound equation. + 10 + 10 - 2 - 2 Solve for x using equation rules. x = 10 or x = - 2 These are the possible solutions to our equation. 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 -2 solution does not work, leaving us with x = 10 as the key number of paces. The treasure lies 10 paces north and 2x + 4 = 2(10) + 4 = 24 paces east of Leaning Rock, or 2x + 6 = 2(10) + 6 = 26 paces straight from the rock. I have come to realize that the most profound thing I will ever do in my life is this so called Elementary Algebra, this class has been anything but Elementary. The best thing about this class has been having Instructor Tsareva. Its not everyday that you have an instructor that understands her students and care. Instructor Tsareva, thank you for allowing us to learn and apply what you have taught us in this class, you have been by far, the best instructor I have had at Ashford University.

Reference
Dugopolski, M. (2012). Elementary and intermediate algebra (4th ed.). New York, NY:
McGraw-Hill Publishing.