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) -># the bigger number
#-(2x+3)-># the smaller 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
#-(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())#