Answer:
- -2.94829122872 = -90 +3√842
- -177.05170877128 = -90 -3√842
Explanation:
You want two numbers that have a product of 522 and a sum of -180.
Equations
The requirements can be written as the equations ...
x + y = -180
xy = 522
Solution
The first can be used to write an expression for y:
y = -180 -x
Substituted into the second equation, we have ...
x(-180 -x) = 522
x² +180x = -522 . . . . . eliminate parentheses, multiply by -1
x² +180x +8100 = 8100 -522 . . . . . complete the square
(x +90)² = 7578
x = -90 ± 3√842 . . . . . take the square root, add -90
y = -x -180 = -90 ± 3√842 . . . . . . sign opposite that of x
The two numbers are irrational:
- -2.94829122872 = -90 +3√842
- -177.05170877128 = -90 -3√842
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Additional comment
Factor pairs of 522 are ...
522 = 1×522 = 2×261 = 3×174 = 6×87 = 9×58 = 18×29
Sums of these pairs are 523, 263, 177, 93, 67, 47. The closest sum to 180 is 177 = 174 + 3.
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