SOLUTION: what two numbers multiply to -16 and add to 15

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Question 1210241: what two numbers multiply to -16 and add to 15
Found 3 solutions by ikleyn, MathLover1, math_tutor2020:
Answer by ikleyn(52776) About Me  (Show Source):
You can put this solution on YOUR website!
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what two numbers multiply to -16 and add to 15
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Answer. The numbers are 16 and -1.

Here are two ways to solve it. 

First way is to guess it mentally. Usually, it requires several seconds and does not require writing.


Another way is to introduce x for one number, 15-x for another number,
and then to write and to solve this equation for the product 

    x*(15-x) = -16,

    15x - x^2 = - 16,

    x^2 - 15x - 16 = 0,

    (x-16)*(x+1) = 0.      <<<---===  factoring


It gives the answer algebraically:  the numbers are 16 and -1.

Solved.



Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!


let two numbers be x and y

if two numbers multiply to -16, we have
x%2Ay=-16..........solve for x
x=-16%2Fy............eq.1


if two numbers add to 15, we have
x%2By=15.........eq.2..............substitute x+from eq.1
-16%2Fy%2By=15.........multiply by+y
-16%2By%5E2=15y
y%5E2-15y-16=0......factor
y%5E2%2By-16y-16=0
%28y%5E2%2By%29-%2816y%2B16%29=0
y%28y%2B1%29-16%28y%2B1%29=0
%28y+-+16%29+%28y+%2B+1%29+=+0

solutions:
y=16 or y=-1

then
x=-16%2Fy=-16%2F16=-1
or
x=-16%2F-1=16


answer:
x=-1
y=16

or
x=16
y=-1

Answer by math_tutor2020(3816) About Me  (Show Source):
You can put this solution on YOUR website!

You can guess-and-check your way toward the answer.
Here are all of the ways to multiply to -16 when using integers only:
-1*16 = -16
-2*8 = -16
-4*4 = -16
1*(-16) = -16
2*(-8) = -16
Of that list, we see that -1 + 16 = 15.
Therefore the two mystery numbers are -1 and 16.

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Another approach is to follow these steps.
x = one of the mystery numbers
15-x = the other mystery number
Note that x and 15-x add to 15.
They must also multiply to -16, so,
first*second = -16
x*(15-x) = -16
-x^2+15x = -16
-x^2+15x+16 = 0
x^2-15x-16 = 0
You might be thinking "at this point I can factor".
If you used a guess-and-check method to factor, then you basically repeat the same style of steps as the first paragraph shown above.
If you're curious, or want to check your scratch work, then x^2-15x-16 factors to (x+1)(x-16).


Instead of factoring, let's use the quadratic formula.
Recall the template ax^2 + bx + c = 0.
In this case,
a = 1, b = -15, c = -16
which means,
x+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29

x+=+%28-%28-15%29%2B-sqrt%28%28-15%29%5E2-4%281%29%28-16%29%29%29%2F%282%281%29%29

x+=+%2815%2B-sqrt%28225+%2B+64%29%29%2F%282%29

x+=+%2815%2B-sqrt%28289%29%29%2F%282%29

x+=+%2815%2B-++17%29%2F%282%29

x+=+%2815%2B17%29%2F%282%29 or x+=+%2815-17%29%2F%282%29

x+=+%2832%29%2F%282%29 or x+=+%28-2%29%2F%282%29

x+=+16 or x+=+-1
We arrive at the same pair of values found earlier.
If x = 16, then 15-x = 15-16 = -1
If x = -1, then 15-x = 15-(-1) = 16

The quadratic formula route is useful when you have larger numbers to work with.
For example, let's say the problem was "Find two numbers that multiply to 476 and add to 45".
To solve this example problem, you would need to solve x*(45-x) = 476.
Rearranging things leads to x^2-45x+476 = 0.
The guess-and-check route might take a while (and it's possible to overlook some cases).
The quadratic formula is preferable here. Using that formula yields x = 17 and x = 28 as the two solutions
17*28 = 476
17+28 = 45


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Answer: -1 and 16
-1*16 = -16
-1+16 = 15