SOLUTION: What multiplies to give you -60 and adds to give you -1

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Question 854758: What multiplies to give you -60 and adds to give you -1
Found 2 solutions by Fombitz, Seutip:
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Let the two numbers be X and Y.
1.X%2AY=-60
2.X%2BY=-1
From eq. 2,
X=-1-Y=-%281%2BY%29
Substitute into eq. 1,
-%281%2BY%29Y=-60
Y%2BY%5E2=60
Y%5E2%2BY-60=0
Use the quadratic formula,
Y+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
Y+=+%28-1+%2B-+sqrt%28+1%5E2-4%2A1%2A%28-60%29+%29%29%2F%282%2A1%29+
Y+=+%28-1+%2B-+sqrt%28+1%2B240+%29%29%2F%282%29+
Y+=+%28-1+%2B-+sqrt%28+241+%29%29%2F%282%29+
So,
when Y+=+%28-1+%2B+sqrt%28+241+%29%29%2F%282%29+,
X=-1-+%28-1+%2B+sqrt%28+241+%29%29%2F%282%29+
X=-1%2B1%2F2-sqrt%28241%29%2F2
X=-1%2F2-sqrt%28241%29%2F2
and
when Y+=+%28-1+-+sqrt%28+241+%29%29%2F%282%29+,
X=-1-+%28-1+-+sqrt%28+241+%29%29%2F%282%29+
X=-1%2B1%2F2%2Bsqrt%28241%29%2F2
X=-1%2F2%2Bsqrt%28241%29%2F2

Answer by Seutip(231) About Me  (Show Source):
You can put this solution on YOUR website!
Let a - be one of the number
b - be the second number
When given a question of this nature, you can look upon the two numbers as the factors in two polynomials that give you a quadratic when multiplied. That sounds complicated, but I mean this:
%28x+%2B+a%29%28x+%2B+b%29+
Use our friend Mr. Foil
x%5E2+%2B+%28a+%2B+b%29x+%2B+ab+
So they give you a product of "ab" and a sum of "a + b". You are looking for exactly that, two numbers that multiply to give a product you specify and add to give the sum you specify.
If you substitute the sum, -1, and the product, -60, you get:
x%5E2%2B%28-1%29x%2B%28-60%29-0
x%5E2-x-60=0 Since it's not a perfect trinomial we use completing the square
x%5E2-x%2B1%2F4=60%2B1%2F4 It's 1/4 because 3rd term = ((-1)/2)^2
sqrt%28x-1%2F2%29%5E2=sqrt%2860%2A1%2F4%29
x-1%2F2=sqrt15
x=sqrt15%2B1%2F2

Hope that helps!