MHB Digit word problems -linear equation

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The discussion centers on solving a problem involving two consecutive positive odd integers where the difference of their squares equals 64. The first solution incorrectly calculates the value of x, yielding integers 19 and 21 instead of the correct pair. The second solution correctly identifies the integers as 15 and 17, with x equating to 15. The confusion arises from the miscalculation in the first method, where x should have been 7 instead of 9. Ultimately, the second method provides the accurate answer to the problem.
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can you please tell why my solutions here yield different answers

here's the problem,

Find two consecutive positive odd integers such that the difference of their squares is 64.

my first solution is

let $2x+1=$ smaller odd interger
$2x+3=$ larger odd integer

$(2x+3)^2-(2x+1)^2=64$

$x=9$

the numbers are 19 and 21.

my 2nd solution

let $x=$ smaller odd interger
$x+2=$ larger odd integer

$(x+2)^2-(x)^2=64$

$x=15$

the numbers are 15 and 17.

which solution method is correct?

thanks!
 
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paulmdrdo said:
$(2x+3)^2-(2x+1)^2=64$

$x=9$
This should be $x=7$.
 
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