Question

Question #d3b12

Answer 1

Two Numbers: 13 and -5

Explanation:

The pairs of numbers that have a product of -65:
`{-1,65},{1,-65},{-5,13},{-13,5}`

By summing each of these pairs we see that {-5,13} has the product of -65 and sum of 8

Answer 2

See a solution process below:

Explanation:

First, let's call the two numbers: `n` and `m`

From the first statement in the problem we can write this equation:

`n xx m = -65`

From the second statement in the problem we can write this equation:

`n + m = 8`

Step 1) Solve the second equation for `n`:

`n + m = 8`

`n + m - color(red)(m) = 8 - color(red)(m)`

`n + 0 = 8 - color(red)(m)`

`n = 8 - m`

Step 2) Substitute `(8 - m)` for `n` in the first equation and solve for `m`:

`n xx m = -65` becomes:

`(8 - m) xx m = -65`

`8m - m^2 = -65`

`8m - color(blue)(8m) + color(red)(m^2) - m^2 = color(red)(m^2) - color(blue)(8m) - 65`

`0 + 0 = m^2 - 8m - 65`

`m^2 - 8m - 65 = 0`

`(m - 13)(m + 5) = 0`

Solution 1:

`m - 13 = 0`

`m - 13 + color(red)(13) = 0 + color(red)(13)`

`m - 0 = 13`

`m = 13`

Solution 2:

`m + 5 = 0`

`m + 5 - color(red)(5) = 0 - color(red)(5)`

`m + 0 = -5`

`m = -5`

The two numbers are: `13` and `-5`

`13 + -5 = 8`

`13 xx -5 = -65`

Answer 3

The two numbers are -5 and 13.

Explanation:

Let the first number be set to `x`.
Let the second number set to `y`.
`xy`=-65
`x+y=8`
`y=8-x`
`xy`
`=x(8-x)`
`8x-x^2=-65`
`x^2-8x-65=0`
`(x-13)(x+5)=0`
`x=13` or `x=-5`
Therefore
`y=-5 if x=13`
Since `x` and `y` are just defined, they are interchangable.
So, the two numbers are 5 and -13.
`Q.E.D.`