I found a really interesting problem with a nice solution.


Find the number of real solutions to this equation.



I may have remembered the problem wrong  ;-;

 Jun 19, 2024

We can't go ahead and directly solve for x, so let's take a different approach. 

Let's first set \(u=\sqrt[3]{x+14}\). We can solve for u, then plug it in to solve for x. 

For this value of u, we get \(x=u^3-14\)


Subsituting out all x, we get that 



Removing the roots, we have 

\(-u^3+28=4-3\cdot \:4^{\frac{2}{3}}u+3\sqrt[3]{4}u^2-u^3\)


Combining all like terms, we ge tthat

\(3\sqrt[3]{4}u^2-3\cdot \:4^{\frac{2}{3}}u-24=0\)


Using the quadratic equation, we have



Now subbing this value of u and finding x, we have

\(\sqrt[3]{x+14}=\frac{4^{\frac{2}{3}}+6\sqrt[3]{2}}{2\sqrt[3]{4}} \\ x+14=\frac{55}{2}+\frac{9}{2}\\ x=18\)



\(\sqrt[3]{x+14}=\frac{4^{\frac{2}{3}}-6\sqrt[3]{2}}{2\sqrt[3]{4}}\\ x+14=\frac{1}{2}-\frac{9}{2}\\ x=-18\)


So our answers are 18 and -18. 

I completed this in a really complicated manner. I'm interested to see if there was a more efficient way to complete this.


Thanks! :)

 Jun 19, 2024

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