SOLUTION: The difference of two numbers is 2 and the sum of their square twenty what are the numbers

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Question 990548: The difference of two numbers is 2 and the sum of their square twenty what are the numbers

Found 2 solutions by ikleyn, macston:
Answer by ikleyn(52778) About Me  (Show Source):
You can put this solution on YOUR website!
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Answer.  a)  4  and  2;   b)  -2  and  -4.

Solution

x - y = 2;   ----->

%28x-y%29%5E2 = 4,   ----->

x%5E2+-+2xy+%2B+y%5E2 = 4,   (recall that   x%5E2+%2B+y%5E2+=+20) ----->

20+-+2xy = 4, ----->

2xy = 20 - 4 = 16, ----->

xy = 16%2F2 = 8. ----->

system%28x-y=2%2C%0D%0Axy+=+8%29. ----->

x = 2+y ----->

(2+y)*y = 8, ----->

y%5E2+%2B+2y+-+8 = 0 ----->

The roots are   y%5B1%5D = 2,   y%5B2%5D = -4   (use the quadratic formula or the Viete's theorem). ----->

The solutions are   (x,y) = (4.2)   and   (x,y) = (-2, -4).


Answer by macston(5194) About Me  (Show Source):
You can put this solution on YOUR website!
.
x and y are the two numbers
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x-y=2 or y-x=2 (order is not specified)
x=2+y or x=2-y
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x%5E2%2By%5E2=20
%282%2By%29%5E2%2By%5E2=20
%28y%5E2%2B4y%2B4%29%2By%5E2=20
2y%5E2%2B4y%2B4=20
2y%5E2%2B4y-16=0
y%5E2%2B2y-8=0
%28y%2B4%29%28y-2%29=0
y%2B4=0 OR y-2=0
y=-4 OR y=2
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For y=-4:
x=y+2=-4+2=-2
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CHECK:
x^2+y^2=20
(-4)^2+(-2)^2=20
16+4=20
20=20
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For the first solution the difference between first and second:
x-y=2
-2-(-4)=2
-2+4=2
2=2
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ANSWER 1: One solution is (-2,-4)
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For y=2:
x=y+2=2+2=4
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CHECK:
x^2+y^2=20
2^2+4^2=20
4+16=20
20=20
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For second solution the difference between second and first:
y-x=2
4-2=2
2=2
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ANSWER 2: Another solution is 2,4