the product of two consecutive negative integers is 240. what are the integers?

Question

I already know that it's n= -16 and 15 but l don't know how to do the proof/checking if my answer was correct. 

Answer 1

240=2*2*2*2*3*5=(2*2*2*2)*(3*5)=16*15

-15 and -16 are the numbers

Answer 2

Hi Crypt,

 

The fact that you have 2 negative integers is thankfully irrelevant, we can do this problem just like we have two positive integers and slap negative signs on them at the end. This is because, as you know, the result of multiplying two negative integers together is the same as multiplying two positive integers together.

 

So the key condition is that the integers are consecutive. Algebraically, that means if our first integer is n, the other integer is n+1. That means:

 

n x (n+1) = 240

n² + n = 240

n² + n - 240 = 0

 

Now at this point, you can solve for n with either factoring, or the big long quadratic formula (both will give you the same results). Factoring is going to be a bit harder for this one, since that depends on you knowing ahead of time that 16 x 15 = 240, and that's kind of the whole point of the problem. So let's try the quadratic equation instead.

 

For any equation with the format Ax² + Bx + C, the quadratic formula says x can have 2 values, given by:

 

x = [-B + squareroot(B² - 4AC)] / 2A

 

OR

 

x = [-B - squareroot(B2 - 4AC)] / 2A

 

So for us, our equation is n² + n - 240 = 0. That means, when it comes to using the Quadratic Formula to solve for n our values for x, A, B, and C are as follows:

 

x = n

A = 1

B = 1

C = -240

 

Now all we do is plug our values for A, B, and C into that quadratic formula mess:

 

x = [-B + squareroot(B² - 4AC)] / 2A

n = [-1 + squareroot(1² - {4 x 1 x -240} )] / (2 x 1)

n = [-1 + squareroot(1 - {-960} )] / 2

n = [-1 + squareroot(1 + 960)] / 2

n = [-1 + squareroot(961)] / 2

n = [-1 + 31] / 2

n = 30 / 2

n = 15

 

Technically there's still the other half of the quadratic formula we need to do; above, we only did 

 

x = [-B + squareroot(B² - 4AC)] / 2A,

 

but remember that there's also

 

x = [-B - squareroot(B² - 4AC)] / 2A

 

as well.

 

However, the problem says that we're supposed to find 2 consecutive negative integers that multiply together to form 240, and we just found what one of them is. We don't need to do the quadratic formula again to find the other integer - all we need to do is figure out what consecutive number to multiply to 15 to get 240.

 

That means the other number we're looking for is either 14 or 16 (those are the only integers consecutive with 15). 15 x14 = 210, so it's not what we're looking for. 15 x 16 = 240, so bingo! So one of the numbers is 15, and the other is 16.

 

Since the problem said the 2 integers are consecutive AND negative, all we need to do is stick negative signs on both 15 and 16. To check our work, we see that -15 x -16 = 240, so we're all good!

 

The answers are therefore -15 and -16.

Answer 3

Let n = the smaller integer

Let n + 1 = the larger integer

n(n + 1) = 240

n² + n = 240

n² + n - 240 = 0

(n + 16)(n - 15) = 0

n + 16 = 0                n - 15 = 0

n = -16                     n = 15, not a negative integer

n + 1 = 0

16 + 1 = -15 

The two negative answers are -16 and -15.  To check, multiply them together.