Question 112422
If the smaller of the two consecutive integers is x, then the other one must be x + 1.


The product of these two integers is then:  x(x+1), and the sum of these two integers is x + (x + 1), or 2x + 1.


Now we can translate the first sentence of the problem into mathematical symbols:


The product of two consecutive integers (x(x + 1)) is (=) 9 times their sum (9(2x + 1)), less 9 (-9).  Or,


{{{x(x+1)=(9(2x+1))-9}}}


Now simplify and solve:


{{{x^2+x=18x+9-9}}}
{{{x^2+x-18x=0}}}
{{{x^2-17x=0}}}
{{{x(x-17)=0}}}, and finally,
{{{x=0}}} or {{{x=17}}}.


This means that the two consecutive integers are either 0 and 1 or 17 and 18.


Check:
{{{0*(0+1)=9*(2*0+1)-9}}}
{{{0=0}}}, and our first answer checks.


{{{17*18=9*(17+18)-9}}}
{{{306=9*(35)-9}}}
{{{306=315-9=306}}}, and the second answer checks also.