SOLUTION: the squarqe of the sum of two numbers is 144 and the sum of their squares is 80. find the number

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Question 1133746: the squarqe of the sum of two numbers is 144 and the sum of their squares is 80. find the number
Found 2 solutions by Edwin McCravy, Alan3354:
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
system%28%28x%2By%29%5E2=144%2Cx%5E2%2By%5E2=80%29

system%28x%5E2%2B2xy%2By%5E2=144%2Cx%5E2%2By%5E2=80%29

Subtract the two equations:

%28x%5E2%2B2xy%2By%5E2%29-%28x%5E2%2By%5E2%29=144-80

x%5E2%2B2xy%2By%5E2-x%5E2-y%5E2=64

2xy=64

xy=32

y=32%2Fx

Substitute in the original equation:

x%5E2%2By%5E2=80

x%5E2%2B%2832%2Fx%29%5E2=80

x%5E2%2B1024%2Fx%5E2=80

x%5E4%2B1024=80x%5E2

x%5E4-80x%5E2%2B1024=0

%28x%5E2-16%29%28x%5E2-64%29=0

%28x-4%29%28x%2B4%29%28x-8%29%28x%2B8%29=0

x-4=0; x+4=0; x-8=0; x+8=0
  x=4;   x=-4;  x=8;   x=-8

Substitute each in {{y=32/x}}}

y=32%2F4;y=32%2F%28-4%29; y=32%2F8; y=32%2F%28-8%29

y=8;y=-8; y=4; y=-4

Solutions are

(x,y)=(4,8), (-4,-8), (8,4), (-8,-4)

Edwin

Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
the squarqe of the sum of two numbers is 144 and the sum of their squares is 80. find the number
===============
(x + y)^2 = 144
x^2 + y^2 = 80
----
x + y = 12 ---> y = 12 - x
x^2 + y^2 = 80
x^2 + (12-x)^2 = 80
2x^2 - 24x + 64 = 0
x^2 - 12x + 32 = 0
x = 4, x = 8
y = 8, y = 4
============================
x + y = -12 ---> y = -12 - x
x^2 + y^2 = 80
x^2 + (-12-x)^2 = 80
2x^2 + 24x + 64 = 0
x^2 + 12x + 32 = 0
x = -4, x = -8
y = -8, y = -4
-------------
One number is + or -4, the other is + or -8,
and the absolute value of the sum is 12.