Question 338136
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The first thing to do is draw yourself a sketch.  You end up with a right triangle with the line from Karen's house to Dan's house being the hypotenuse and the restaurant at the right angle vertex.


Then there are two ways to go about this.  You can do it the easy way in your head if you say the hypotenuse is 10 and if the triangle were a 3-4-5 right triangle, the two legs would be 6 and 8.  And since 6 and 8 differ by 2, Karen's house is 8 miles from the restaurant.


Or you can do it the hard way.


Let *[tex \Large x] represent the distance from Karen's house to the restaurant.  Then the distance from Dan's house to the restaurant is *[tex \Large x\ -\ 2]


Then Pythagoras:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x^2\ +\ (x\ -\ 2)^2\ =\ 100]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x^2\ +\ x^2\ -\ 4x\ +\ 4\ =\ 100]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 2x^2\ -\ 4x\ -\ 96\ =\ 0]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ x^2\ -\ 2x\ -\ 48\ =\ 0]


You can take it from here.  Toss the extraneous negative root.



John
*[tex \LARGE e^{i\pi} + 1 = 0]
My calculator said it, I believe it, that settles it
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