# Solving the factorial equation $(n + 4)! = 90(n + 2)!$

Solve the equation below:

$(n+4)! = 90 (n+2)!$

I did this:

$(n+4)(n+3)(n+2)! = 90 (n+2)!$

$n^2+7n+12+90=0$

$n^2+7n+102=0$

Is there anymore to this?

• You made a sign error with the $+90$. If you correct that mistake you will find the equation easier to solve.
– user575517
Sep 18, 2018 at 15:51
• "Is there anymore to this?" No. There isn't. ... except you have a sign error. $(n+4)(n+3) = 90$ so $n^2 + 7n + 12 -90 = n^2 + 7n -78=0$. Okay, there's a little more. Then answer has to be a positive integer and only one of the two roots can be positive. Sep 18, 2018 at 16:31

When you take the $90$ to the other side you need to subtract it. You'll get

$n^2 +7n +12 -90 = 0,$

$n^2 +7n -78 = 0,$

$(n - 6)(n+13) = 0,$

So $n = 6$ or $n = -13$.

Clearly $n = -13$ doesn't make sense, so we conclude:

$n = 6$.

Well, you know that $$\frac{(n+4)!}{(n+2)!} = (n+4)\cdot(n+3) = n^2+7n+12=90$$$$n^2+7n-78=0$$The solutions to this quadratic are $n=6$ and $n=-13$, so we take $n=6$ to be our answer.

• @N.F.Taussig Thank you! That's silly of me Sep 18, 2018 at 16:27

You can go to the quadratic if you wish, and that solves the problem. But if you simply divide through by $(n+2)!$ you obtain $$(n+3)(n+4)=90$$ and it is easy to see the factorisation of $90$ into two consecutive positive integer factors $(9\times 10)$. The quadratic method does make sure you don't miss the alternative (irrelevant here) $90=-10\times -9$. The fact that it is a quadratic means that you know that there are no more solutions.

• Worth noting that "consecutive" means $b = a+1$ which means $a \approx b$ which means $a*b = 90$ and $a*b \approx a^2$ so $a\approx b \approx \sqrt {90}$ and as $a$ and $b$ are consecutive it follows that if there is a solution it is $n + 3 = \lfloor \sqrt {90}\rfloor = 9$ and $n+4 = \lceil \sqrt {90} \rceil =10$ or $n = 6$. Sep 18, 2018 at 16:37
• @fleablood good point! Sep 18, 2018 at 18:44
• It didn't occur to me to do simple factoring. I went straight to the solving the quadratic.... Sep 18, 2018 at 19:12

Just to be different (and to rip off Mark Bennett's answer):

$ab = k> 0$ with $0< a \le b$ means $a \le \sqrt {k}$ and $b \ge \sqrt {k}$. (and if $0< a < b$ then $a < \sqrt k$ and $b > \sqrt{k}$)

So $(n+3)(n+4) = 90$ means $n+3 \le \sqrt{90} \approx 9.smallchange \le n+4$

As $n+3$ and $n+4$ are consecutive we must have:

$(n+3) \le \sqrt{90} < (n+3) + 1$.

Or in other words $n + 3 = \lfloor \sqrt{90} \rfloor = 9$.

So $n = 6$.