If the product of two consecutive integers is decreased by 20 times the greater integer, the result is 442. find the integers

Let the first integer be x. Then the next consecutive integer is x+1.

Translating the word problem into symbols, x (x+1) - 20 (x+1) = 442.

Then x^2 + x - 20x - 20 = 442

or x^2 - 19x - 462 = 0

Solve by completing the square:

x^2 - 19x + 90.25 - 90.25 - 462 = 0

(x-9.5) ^2 - 552.25 = 0

Taking the positive square root of both sides, we get x-9.5 = 23.5, or

x = 33

The two consecutive integers are 33 and 34.

Check: (33) (34) - 20 (34) = 442 (as expected).