Pythagorean Theorem
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?
Solving for x, the solution involves plugging in the binomials mentioned in the problem using the Pythagorean theorem. The Pythagorean theorem states that in a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. The Pythagorean theorem is expressed in the following equation: a2 + b2 = c2
The problem gives information about the other two sides; in finding "x," both Ahmed and Vanessa can triangulate the location of the buried treasure.
To start, plug in the binomials given in the problem into the Pythagorean theorem equation. Let a=x, b=2x+4, and c=2x+6:
(x)2 + (2x+4)2 = (2x+6)2
Computing for the squared binomials, x2 + 4x2 +16x +16 = 4x2 +24x + 36
Completely solving both sides of the equation to arrive at the final quadratic equation by transposing the right side of the equation to the left side of it:
X2 + 4x2 -4x2 +16x -24x +16 -36 = 0
X2 -8x -20 = 0
To solve the quadratic equation, zero factor will be used. The coefficient is negative, there will be positive and negative factors. Two factors of -20 should add up to -8. 10 and 2 are factors of 20 that would also add up to 8 if the greater factor is negative and lower factor, positive. The result are two binomials to be solved using zero factor.
(x -- 10) (x + 2) = 0
After solving the binomials, the result is a compound equation expressed as follows (also the possible solutions for the problem):
x = 10 or x = -2
Considering the values of x solved through Pythagoras theorem, x=10 is more plausible given the buried treasure problem (negative distance is not applicable in the problem presented).
Computing for the binomials at x=10:
The treasure lies 10 paces to the north, and 2(10)+4=24 paces to the east. Or, the treasure is buried 2(20)+6=26 paces from Castle Rock.
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