The product of two consecutive positive integers is 75 more

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

Solution

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.