document.write( "Question 539770: If a number is increased or decreased by 10, you will have a square number what is the number? \n" ); document.write( "

Algebra.Com's Answer #353548 by KMST(5328) $\"\"$ $\"About$

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If a number, n, is increased by 10, you get n+10.
\n" ); document.write( "If the same number, n, is decreased by 10, you get n-10.
\n" ); document.write( "They are both perfect squares.
\n" ); document.write( "The difference between then is (n+10)-(n-10)=n+10-n+10=20.
\n" ); document.write( "That tells you they are rather small numbers because as numbers grow, the differences between the squares of consecutive numbers grow larger and larger, and it does not take much to go past 20.
\n" ); document.write( "TRIAL AND ERROR
\n" ); document.write( "You could make a list of squares (of numbers 1 through 10), and look for two squares that differ by 20.
\n" ); document.write( "ANOTHER WAY
\n" ); document.write( "Could n-10 and n+10 be the squares of consecutive numbers x, and x+1?
\n" ); document.write( "Would they be the squares of numbers differing by 2, like y and y+2?
\n" ); document.write( " $\"%28x%2B1%29%5E2-x%5E2=x%5E2%2B2x%2B1-x%5E2=2x%2B1\"$ will be odd and could never equal 20. So it's not squares of consecutive numbers.
\n" ); document.write( " $\"%28y%2B2%29%5E2-y%5E2=y%5E2%2B4y%2B4-y%5E2=4y%2B4=4%28y%2B1%29\"$ will be 20 when $\"y=4\"$
\n" ); document.write( "So we are talking about the squares of $\"y=4\"$ and $\"y%2B2=4%2B2=6\"$
\n" ); document.write( " $\"4%5E2=16=n-10\"$ and $\"%284%2B2%29%5E2=6%5E2=36=n%2B10\"$
\n" ); document.write( "Solving either $\"16=n-10\"$ and/or $\"36=n%2B10\"$ will give you $\"n=26\"$ \n" ); document.write( "

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