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Devin
JordanI think you probably want it the other way around: numbers that add to -9 and multiply to 20. These would be -4 and -5.
How you wrote it:
x + y = 20
xy = -9
x = 20 - y
(20 - y)y = -9
-y^2 + 20y = -9
y^2 - 20y - 9 = 0
(-(-20) +/- sqrt((-20)^2 - 4(1)(-9)))/(2(1)) = y
(20 +/- sqrt(436))/2
10 +/- 2sqrt(109)/2
10 +/- sqrt(109) = y
x + y = 20
x + (10 +/- sqrt(109) = 20
If +,
x + 10 + sqrt(109) = 20
x = 10 - sqrt(109)
If -,
x + 10 - sqrt(109) = 20
x = 10 + sqrt(109)
The two numbers are thus:
10 + sqrt(109) and 10 - sqrt(109)
Check:
x + y = 20
(10 + sqrt(109)) + (10 - sqrt(109)) = 20
10 + 10 + sqrt(109) - sqrt(109) = 20
20 = 20
xy = -9
(10 + sqrt(109))(10 - sqrt(109)) = -9
Difference of squares => (a+b)(a-b)=(a^2 - b^2)
10^2 - sqrt(109)^2 = -9
100 - 109 = -9
-9 = -9
The solutions do work, so the numbers are:
10 + sqrt(109) and 10 - sqrt(109)
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