The sum of a positive integer and its square is 90. What is the number?

1 Answer
sente
Dec 27, 2016

#9#

Explanation:

Let #n# be the integer in question. Then we have

#n^2+n = 90#

#=> n^2+n-90 = 0#

We now have a quadratic equation to solve. We could use the quadratic formula, however we know that #n# is an integer, so instead let's try to solve by factoring instead.

#n^2+n-90 = 0#

#=> n^2 + 10n - 9n - 90 = 0#

#=> n(n+10)-9(n+10) = 0#

#=> (n-9)(n+10) = 0#

#=> n-9 = 0 or n+10 = 0#

#=> n=9 or n=-10#

As it is given that #n>0#, we can disregard the possibility that #n=-10#, leaving us with our final answer of #n=9#

Checking our result, we find that it satisfies the given conditions:

#9+9^2 = 9+81 = 90#

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