SOLUTION: The difference between two positive nubers is 5 and the sum of their squares is 233. What are the numbers?

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Question 123198This question is from textbook
: The difference between two positive nubers is 5 and the sum of their squares is 233. What are the numbers? This question is from textbook

Answer by MathLover1(20850) About Me  (Show Source):
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The difference between two positive nubers is 5
if numbers are x and y where x%3Ey, then:
x-y=5
and the sum of their squares is 233
x%5E2+%2B+y%5E2+=+233


What are the numbers
x-y=5......=>.....x=y%2B5}
substitute x in x%5E2+%2B+y%5E2+=+233
%28y%2B5%29%5E2+%2B+y%5E2+=+233
y%5E2+%2B+10y+%2B+25+%2B+y%5E2+=+233
2y%5E2+%2B+10y+%2B+25+-+233+=+0

2y%5E2+%2B+10y+-+208=+0
y%5E2+%2B+5y+-+104+=+0

y%5B1%2C2%5D+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
y%5B1%2C2%5D+=+%28-5+%2B-+sqrt%28+5%5E2+-+4%2A1%2A%28-104%29+%29%29%2F%282%2A1%29+
y%5B1%2C2%5D+=+%28-5+%2B-+sqrt%28+25+%2B+416+%29%29%2F2+
y%5B1%2C2%5D+=+%28-5+%2B-+sqrt%28+441%29%29%2F2+
y%5B1%2C2%5D+=+%28-5+%2B-+21%29%2F2+
y%5B1%5D+=+%28-5+%2B+21%29%2F2+
y%5B1%5D+=+16%2F2+
y%5B1%5D+=+8+

y%5B2%5D+=+%28-5+-+21%29%2F2+
y%5B2%5D+=+%28+-26%29%2F2+
y%5B2%5D+=++-13


......=>.....x1=y1%2B5=+8%2B5=13}

......=>.....x2=y2%2B5=+-13%2B5=-8}

check:
x-y=5......=>.....13-8=5
and -8-%28-13%29=-8%2B13=5

13%5E2+%2B+8%5E2+=+233
169+%2B+64+=+233
233=+233
and
%28-8%29%5E2+%2B+%28-13%29%5E2+=+233
64+%2B+169++=+233
233=+233