# 2 consecutive odd negative integers have a product of 399. What are the integers?

##### 1 Answer

#### Explanation:

You know that you're looking for **consecutive negative integers**, so right from the start you should expect the two two numbers to take the form

#-(2x+1) -># thebigger number

#-(2x+3)-># thesmaller number

This is the case because a *positive* odd integer can be expressed as

#2x + 1#

where

The consecutive *even* number would be

#(2x+1) + 1 = 2x+2#

which makes the consecutive **odd** number

#(2x+1) + 2 = 2x+3#

Since the two numbers are **negative**, all you have to do is tag along a minus sing.

So, you know that

#-(2x+1) * [-(2x+3)] = 399#

Expand to get

#4x^2 + 6x + 2x + 3 = 399#

Rearrange to quadratic equation form

#4x^2 + 8x -396 = 0#

Now, this quadratic equation has two possible solutions, as given by the **quadratic formula**

#x_(1,2) = (-8 +- sqrt( 8^2 - 4 * 4 * (-396)))/(2 * 4)#

#x_(1,2) = (-8 +- sqrt(6400))/8#

#x_(1,2) = (-8 +- 80)/8 implies {(x_1 = (-8 +80)/8 = 9), (x_2 = (-8 - 80)/8 = -11) :}#

You need **negative**, which means that your two consecutive odd integers will be

#-(2 * 9+1) = -19" "# and#" "-(2 * 9 + 3) = -21#

Do a quick check to make sure that the calculations are correct

#-19 * (-21) = 399" "color(green)(sqrt())#