The sum of the reciprocals of two consecutive integers is - 9/20. Find the two integers

Suppose the first integer is x, the second one is then x+1

reciprocals are 1/x and 1 / (x+1)

1/x + 1 / (x+1) = - 9/20

make the denominators the same:

1 (x+1) / x (x+1) + 1x/x (x+1) = - 9/20

simplify the demonstrator and add up the numerators: (2x+1) / (x^2+x) = - 9/20

20 (2x+1) = - 9 (x^2+x)

40x+20=-9x^2-9x

9x^2 + 49x+20=0

factor: (1+5) (9x+4) = 0

x=-5 or x=-4/9 (this one doesn't work because it is not an integer)

so the first integer is - 5, the second integer is - 4