SOLUTION: Number that multiplies to -44 but adds to -1

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Question 1210275: Number that multiplies to -44 but adds to -1
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13198) About Me  (Show Source):
You can put this solution on YOUR website!
cross%28Number%29 Numbers that cross%28multiplies%29 multiply to -44 but cross%28adds%29 add to -1.

Trial and error, or any other method, shows that there are not any two rational numbers that satisfy the conditions. So...

a%2Bb=-1 --> b=-a-1

ab=-44
a%28-a-1%29=-44
a%28a%2B1%29=44
a%5E2%2Ba=44
a%5E2%2Ba-44=0

Solve using the quadratic formula.

a=%28-1%2B-sqrt%281%2B4%2A44%29%29%2F2

ANSWER: the two numbers are %28-1%2Bsqrt%28177%29%29%2F2 and %28-1-sqrt%28177%29%29%2F2

Answer by ikleyn(52778) About Me  (Show Source):
You can put this solution on YOUR website!
.

Here is another solution. 

Since the sum of the numbers is -1, it means that the numbers are equally remoted from -0.5,
so we can write

    a = -0.5 + x,

    b = -0.5 - x.


Thus a + b = -1 is provided, and we should satisfy the other condition

    a*b = -44.


It gives

    (-0.5+x)*(-0.5-x) = -44,

    0.5%5E2 - x%5E2 = -44,

    0.025 + 44 = x%5E2

    x%5E2 = 44.25

    x = sqrt%2844.25%29.


Thus,  the numbers are  a = -0.5+%2B+sqrt%2844.25%29 = %28-1+%2B+sqrt%28177%29%29%2F2,  b = -0.5+-+sqrt%2844.25%29 = %28-1+-+sqrt%28177%29%29%2F2.

Solved.

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I posted this solution to surprise you that such simple problem may have two different approaches.
My other goal was to show you a solution which has a geometric meaning - not only a calculation exercise.