The product of two consecutive positive integers is 75 more than nine times the second integer. What are the integers?

Let's note the first integer as x and the secon
consecutive integer is x+1.

Now, we'll write mathematically
the condition of enunciation:

- the product of 2
consecutive integers: x(x+1)

- is:
=

- 75 more: 75 +

- nine times
the second integer: 9(x+1)

Now, let's join
them:

x(x+1) = 75 +
9(x+1)

We'll remove the
brackets:

x^2 + x = 75 + 9x +
9

We'll combine like
terms:

x^2 + x = 84 + 9x

We'll
subtract 84 + 9x both sides:

x^2 + x - 84 - 9x =
0

We'll combine like
terms:

x^2 - 8x - 84 = 0

We'll
apply the quadratic formula:

x1 = [8 + sqrt(64 +
336)]/2

x1 = (8 + 20)/2

x1 =
14

or

x1 =
(8-20)/2

x1 =
-6

Since the integer has to be positive,
we'll accept just x1 = 14.

The
second consecutive integer is x2 =
14+1

x2 =
15.