How Do You Solve (x-1)^2 * (a+x) = 1?

[steem84]

I am a little bit confused about solving the following equation:


(x-1)²(a+x)=1



How to do this??

 

[NoMoreExams]

Are you sure there's a simple way to solve that, given that you don't know a? It's a cubic...

 

[steem84]

well, actually this is the original equation (see figure1)

The solution is in figure 2..


So let me reformulate my question: Can this be proven analytically?

 

[gabbagabbahey]

well, actually this is the original equation (see figure1)

The solution is in figure 2..


So let me reformulate my question: Can this be proven analytically?

This is very different from the problem in your first post!

Anyways, start by squaring both sides of the equation in the first link. Then multiply both sides by $$4((\rho^{*})^2+(x^{*})^2)$$ and simplify to obtain:

$$(\rho^{*})^6+((x^{*})^2-4a^2)(\rho^{*})^4+4a^2(a^2-(x^{*})^2)(\rho^{*})^2=0$$

That should tell you that either $$(\rho^{*})^2=0$$ or

$$(\rho^{*})^4+((x^{*})^2-4a^2)(\rho^{*})^2+4a^2(a^2-(x^{*})^2)=0$$

You can use the quadratic equation to solve the above expression for $$(\rho^{*})^2$$ and then take the square root to obtain the final solution.

 

[steem84]

Yes, ok thank you. It did not cross my mind to substitute a variable for a variable^2