Question 1034425
let the first positive number equal x.
then the second positive number equals x+1.


the sum of the reciprocals of those numbers is equal to 17/12.


you get 1/x + 1/(x+1) = 17/12.


multiply both sides of this equation by x * (x+1) to get:


(x+1) + x = 17/12 * (x+1) * x


simplify to get 2x + 1 = 17/12 * (x^2 + x)


multiply both sides of this equaiton by 12 to get:


12 * (2x + 1) = 17 * (x^2 + x)


simplify to get 24x + 12 = 17x^2 + 17x


subtract the left side of the equation from both sides of the equaiton to get:


0 = 17x^2 + 17x - 24x - 12.


simplify to get 0 = 17x^2 - 7x - 12.


this is the same as 17x^2 - 7x - 12 = 0.


solve this quadratic equation using the quadratic formula to get:


x = (7 + sqrt(865))/34 or x = (7 - sqrt(865))/34


this results in x = 1.070908304 or x = -.6591435982.


x  can't be negative, so x has to be equal to 1.070908304.


this makes x + 1 = 2.070908304.


if you did this correctly, you will get 1/1.070908304 + 1/2.070908304 = 17/12.


evaluate the left side of this equation to get 1.416666667 = 17/12.


multiply both sides of this equation by 12 to get 17 = 17.


this confirms the solution is correct.


your positive consecutive numbers are 1.070908304 and 2.070908304.


these are the same as (7+sqrt(865))/34 and 1 + (7 + sqrt(865))/34.


1 + (7 + sqrt(865))/34 can also be written as (41+sqrt(865))/34.


you can solve the quadratic equation graphically as well.


i used the quadratic formula to get an exact answer.
the graphing software i used gave me an approximate answer.
that graph is shown below:


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