SOLUTION: The sume of the reciprocals of two consecutive odd integers is 20/99. Find the integers. A plane and a truck leave the same town at sunrise and head for a town 450 miles away

Algebra ->  Real-numbers -> SOLUTION: The sume of the reciprocals of two consecutive odd integers is 20/99. Find the integers. A plane and a truck leave the same town at sunrise and head for a town 450 miles away      Log On


   

Question 178957: The sume of the reciprocals of two consecutive odd integers is 20/99. Find the integers.

A plane and a truck leave the same town at sunrise and head for a town 450 miles away. The spead of the plane is 3 times the speed of the truck, and the plane arrives 6 hours ahead of the truck. Find the speed of the plane.

Answer by Mathtut(3670) About Me  (Show Source):
You can put this solution on YOUR website!
let the integers be a and a+2
:
%281%2Fa%29%2B%281%2F%28a%2B2%29%29=20%2F99
:
%28a%2B2%2Ba%29%2Fa%28a%2B2%29=20%2F99
:
%282a%2B2%29%2F%28a%5E2%2B2a%29=20%2F99
:
198a%2B198=20a%5E2%2B40a
:
20a%5E2-158a-198=0
:
highlight%28a=9%29-1.1 is thrown out
:
integers are:system%289%2C11%29
:
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 20x%5E2%2B-158x%2B-198+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-158%29%5E2-4%2A20%2A-198=40804.

Discriminant d=40804 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--158%2B-sqrt%28+40804+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-158%29%2Bsqrt%28+40804+%29%29%2F2%5C20+=+9
x%5B2%5D+=+%28-%28-158%29-sqrt%28+40804+%29%29%2F2%5C20+=+-1.1

Quadratic expression 20x%5E2%2B-158x%2B-198 can be factored:
20x%5E2%2B-158x%2B-198+=+%28x-9%29%2A%28x--1.1%29
Again, the answer is: 9, -1.1. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+20%2Ax%5E2%2B-158%2Ax%2B-198+%29