document.write( "Question 1014284: Find two consecutive odd integers such that their product is 35 more than 8 times their sum. \n" ); document.write( "

Algebra.Com's Answer #630612 by KMST(5328) $\"\"$ $\"About$

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$\"n\"$ = the smaller of the two odd integers
\n" ); document.write( " $\"n%2B2\"$ = the next odd integer
\n" ); document.write( " $\"n%28n%2B2%29\"$ = the product of those two consecutive odd integers
\n" ); document.write( " $\"n%2B%28n%2B2%29\"$ = the sum of those two consecutive odd integers\
\n" ); document.write( "We translate \"their product is 35 more than 8 times their sum\" as
\n" ); document.write( " $\"n%28n%2B2%29=8%28n%2B%28n%2B2%29%29%2B35\"$ .
\n" ); document.write( "That is the equation we have to solve, to find $\"n\"$ (and $\"n%2B2\"$ ).
\n" ); document.write( "
\n" ); document.write( "If we find a solution that is even, or negative, or not an integer at all, we discard it.
\n" ); document.write( "If we find a solution that is an odd integer, that is the answer to the problem.
\n" ); document.write( "If we do not find a solution that is an odd integer, the problem has no solution.
\n" ); document.write( "
\n" ); document.write( " $\"n%28n%2B2%29=8%28n%2B%28n%2B2%29%29%2B35\"$
\n" ); document.write( " $\"n%5E2%2B2n=8%28n%2Bn%2B2%29%2B35\"$
\n" ); document.write( " $\"n%5E2%2B2n=8%28n%2Bn%2B2%29%2B35\"$
\n" ); document.write( " $\"n%5E2%2B2n=8%282n%2B2%29%2B35\"$
\n" ); document.write( " $\"n%5E2%2B2n=16n%2B16%2B35\"$
\n" ); document.write( " $\"n%5E2%2B2n-16n=16%2B35\"$
\n" ); document.write( " $\"n%5E2-14n=51\"$
\n" ); document.write( "That is a quadratic equation that can be solved 3 ways.
\n" ); document.write( "BY COMPLETING THE SQUARE:
\n" ); document.write( " $\"n%5E2-14n=51\"$
\n" ); document.write( " $\"n%5E2-14n%2B49=51%2B49\"$
\n" ); document.write( " $\"%28n-7%29%5E2=100\"$ ---> $\"system%28n-7=10%2C%22or%22%2Cn-7=-10%29\"$ ---> $\"system%28n=10%2B7=17%2C%22or%22%2Cn=-10%2B7=-3%29\"$ ---> $\"highlight%28system%28n=17%2Cn%2B2=19%29%29\"$
\n" ); document.write( "BY FACTORING:
\n" ); document.write( " $\"n%5E2-14n=51\"$
\n" ); document.write( " $\"n%5E2-14n-51=0\"$
\n" ); document.write( " $\"%28n%2B3%29%28n-17%29=0\"$ ---> $\"system%28n=17%2C%22or%22%2Cn=-3%29\"$ ---> $\"highlight%28system%28n=17%2Cn%2B2=19%29%29\"$
\n" ); document.write( "BY USING THE QUADRATIC FORMULA:
\n" ); document.write( " $\"n%5E2-14n=51\"$
\n" ); document.write( " $\"n%5E2-14n-51=0\"$
\n" ); document.write( "The quadratic formula, to find the solution to $\"ax%5E2%2Bbx%2Bc=0\"$ is
\n" ); document.write( " $\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"$ .
\n" ); document.write( "In the case of $\"n%5E2-14n-51=0\"$ , $\"system%28a=1%2Cb=-14%2Cc=-51%29\"$ , and
\n" ); document.write( "

---> $\"system%28x=%2814%2B20%29%2F2=34%2F2=17%2C%22or%22%2Cx=%2814-20%29%2F2=%28-6%29%2F2=-3%29\"$ ---> $\"highlight%28system%28n=17%2Cn%2B2=19%29%29\"$ \n" ); document.write( "

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