SOLUTION: the sum of the squares of two consecutive even integers is 452. find the numbers.

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Question 115744This question is from textbook Algebra, structure and meathod, book 1
: the sum of the squares of two consecutive even integers is 452. find the numbers.
 This question is from textbook Algebra, structure and meathod, book 1

Answer by MathLover1(20849) About Me  (Show Source):
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the sum of the squares of two consecutive even integers is 452. find the numbers.

Let two consecutive even integers be:
x and y+=+x%2B2
the sum of the squares:
x%5E2+%2B+y%5E2+=+452
x%5E2+%2B+%28x%2B2%29%5E2+=+452
x%5E2+%2B+%28x%5E2+%2B+4x+%2B+4%29+=+452
2x%5E2+%2B+4x+%2B+4+=+452……..move 452 to the left to get a standard form of quadratic equation
2x%5E2+%2B+4x+%2B+4+%96+452+=+0……..
2x%5E2+%2B+4x+-+448+=+0…….. divide by 2
x%5E2+%2B+2x+-+224+=+0……..use quadratic formula to solve for x

x%5B1%2C2%5D=%28-b+%2B-+sqrt+%28b%5E2+-4%2Aa%2Ac+%29%29+%2F+%282%2Aa%29
x%5B1%2C2%5D=%28-2+%2B-+sqrt+%282%5E2+-4%2A1%2A%28-224%29+%29%29+%2F+%282%2A1%29
x%5B1%2C2%5D=%28-2+%2B-+sqrt+%284+%2B+896%29%29+%2F+2
x%5B1%2C2%5D=%28-2+%2B-+sqrt+%28900%29%29+%2F+2
x%5B1%2C2%5D=%28-2+%2B-+30%29+%2F+2

x%5B1%5D=%28-2+%2B+30%29+%2F+2
x%5B1%5D=+28+%2F+2
x%5B1%5D=+14

x%5B2%5D=%28-2+-+30%29+%2F+2
x%5B2%5D=+-+32+%2F+2
x%5B2%5D=+-+16


So, if x%5B1%5D+=+14 => y%5B1%5D+=+x%2B2=+14+%2B+2=+16……….1.
and if x%5B2%5D+=+-16 => y%5B2%5D+=+x%2B2=+-16+%2B+2=+-14………2.
check solutions:

1. x%5B1%5D%5E2+%2B+y%5B1%5D%5E2+=+452
14%5E2+%2B+16%5E2+=+452
196+%2B+256+=+452
452+=+452
2. x%5B2%5D%5E2+%2B+y%5B2%5D%5E2+=+452
%28-16%29%5E2+%2B+%28-14%29%5E2+=+452
256+%2B+196+=+452
452+=+452