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Looking to get some clarification on how the following equation was solved for variable $N$,

$$rN-aN(N-b)^2 = 0 $$

The three solutions that were found were,

$$N= 0, b + \sqrt{\frac{r}{a}}, b - \sqrt{\frac{r}{a}}$$

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    $N=0$ is an obvious solution. After simplifying by $N$, you get an easy quadratic equation. – 2017-02-13

5 Answers 5

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You can factor out $N$:

$$N(r-a(N-b)^2)=0.$$

Then $N=0$ is obviously a solution. The other two solutions are found from solving the quadratic equation $$r=a(N-b)^2.$$

There are only three solutions since the original expression is a polynomial of degree three.

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writing your equation in the form $$N(r-a(N-b)^2)=0$$ thus we get $$N=0$$ or $$r-a(N-b)^2=0$$ and from here we get $$\frac{r}{a}=(N-b)^2$$ and if $$\frac{r}{a}\geq 0$$ we have $$\sqrt{\frac{r}{a}}=|N-b|$$ can you proceed?

2

Let's assume $\frac{r}{a} \geq 0$.

The equation $$ rN-aN(N-b)^2 = 0 $$ is equivalent to $$ N (r-a(N-b)^2) = 0 $$ Now, if the product of two numbers is $0$, then one of the two numbers must be $0$. Thus either $N=0$ (first solution) or $$ r-a(N-b)^2=0 $$ This last equation is equivalent to $$ \frac{r}{a} = (N-b)^2 $$ which is equivalent to $$ \pm \sqrt{\frac{r}{a}} = N-b $$ or $$ b \pm \sqrt{\frac{r}{a}} = N $$ Hence $N = b + \sqrt{\frac{r}{a}}$ and $N = b - \sqrt{\frac{r}{a}}$ are two more solutions.

There are no more solutions because what we did was exhaustive.

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    thanks for your solution it was very clear – 2017-02-13
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    @fr14 You're welcome. – 2017-02-13
2

Factoring out an $N$, we have

$$N\left(r-a(N-b)^2\right)=0$$

This is clearly $0$ if $N=0$. Assuming $N\neq0$, we divide by $N$ and obtain

$$r-a(N-b)^2=0\implies r=a(N-b)^2\implies \frac{r}{a}=N^2-2bN+b^2$$

Getting everything on the LHS, we have

$$N^2-2bN+b^2-\frac{r}{a}=0$$

Solving this with the quadratic formula, we get the results that you have.

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Hint:

Observe $$rN-aN(N-b)^2$$ factorizes into $$(N)(r-a(N-b)^2).$$ If this expression is equal to $0,$ by the zero product property, either $N=0,$ or $(r-a(N-b)^2)=0,$ what are the roots to latter case ?