Question

The product of two integers is `-120` and their sum is `2`. How can you find the larger of the two numbers?

Answer 1

Answer `D` would be correct.

Explanation:

Let the two integers be `x` and `y`.

Then

`{(xy = -120), (x + y = 2):}`

I would recommend solving through substitution.

`y = -120/x -> x - 120/x = 2`

`x^2 - 120 = 2x`

`x^2 - 2x - 120 = 0`

You could use factoring, completing the square or quadratic formula to solve. I'll use completing the square.

`x^2 - 2x = 120`

`1(x^2 - 2x + 1 - 1) = 120`

`1(x^2 - 2x+ 1) - 1 = 120`

`x^2 - 2x + 1 = 121`

`(x - 1)^2 = 121`

`x- 1 = +-11`

`x = -11 + 1 and 11 + 1`

`x = -10 and 12`

Resubstitute into the initial equation:

`x + y = 2 -> -10 + y = 2 and 12 + y = 2 -> y = 12 and y = -10`

Therefore, the larger number is `+12`.

Hopefully this helps!

Answer 2

`D) +12`

Explanation:

There are many combinations of numbers which multiply to give `-120`

We know one must be positive and one must be negative.

For example `1 xx-120 = -120," " or -2 xx 60 = -120`

However these values have quite a big sum. (If we think about the absolute value and do not look at the signs)

`=-120 +1 =-119" " and 60+(-2) = 58`

To have a sum of only `+2` means that the two numbers are of a very similar size, with the positive one being slightly bigger than the negative one.

Look for 'middle' factors - ie, those very close to the square root.

`sqrt 120 = 10.95`

Find factors of `120` on either side of `10.95 (~~ 11)`

`10 xx 12 = 120!`

The `12` must be positive and the `10` negative.

`-10xx 12 = -120 " " and" "-10+12 = +2`

The larger one is `+12`