Question

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

Answer 1

`-21` and `-19`

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) ->` the bigger number

`-(2x+3)->` the smaller number

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

`2x + 1`

where `x` is practically any number in `ZZ`.

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 `-(2x+1)` and `-(2x+3)` to be 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())`