Buried treasure. 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?
Pythagorean Quadratic
Here is a treasure hunting problem very similar to the one in the textbook.
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 out 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 doesn't indicate which direction the 2x + 9 paces should go, we can assume that his and Buckwheat's paces should end up in the same place. When we sketch this out on scratch paper we see that it forms a right triangle with 2x + 9 being the length of the hypotenuse, and x and 2x + 6 being the legs of the triangle. Now we know how we can use the Pythagorean Theorem to help solve for x.
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, by putting these measurements into the Theorem equation we have
x2 + (2x + 6)2 = (2x + 9)2 The binomials into the Pyth. Thrm.
x2 + 4x2 + 24x + 36 = 4x2 + 36x + 81 The binomials squared. Notice there is a 4x2
on both sides of the equation which can be
-4x2 -4x2 subtracted out first.
x2 + 24x + 36 = 36x + 81 Subtract 36x from both sides of equation.
-36x -36x
x2 - 12x + 36 = 81 Subtract 81 from both sides of equation.
-81 -81
x2 - 12x - 45 = 0 Now we have a quadratic equation to solve by factoring
and using the zero factor.
(x - ) (x + ) = 0 Since the coefficient of x2 is 1 we can start with a pair of
parenthesis with an x in each. Since the 45 is negative we
know there will be one + and one - in the binomials.
We need two factors of -45 which add up to -12.
-1, 45; -3, 15; -5, 9
1, -45; 3, -15; 5, -9.
Looks like 3, -15 will do it!
(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.
x = 15 or x = -3 These are the possible solutions to our equation.
However, one of these solutions is what we call extraneous because it doesn't 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 as the crow flies!