Solve the equation $4x^3+13x^2-14x=3-\sqrt{15x+9}$ I ran into this problem:
$$4x^3+13x^2-14x=3-\sqrt{15x+9}$$
It makes no sense to square it. I thought it was necessary to make a replacement. What kind of substitution?
First of all, the root gets in the way, so you have to make a substitution that removes the root!
I was able to find such a substitution
$$x=\frac{9}{15}\cos 2t\Rightarrow \sqrt{15x+9}=3\sqrt{2}\cos t$$
We don't have to put the module, because the restrictions allow us to do so
After we got rid of the root, then comes the second problem. How do we solve this equation?
$$\frac{108}{125}\cos^32 t+\frac{117}{25}\cos^22t-\frac{42}{5}\cos 2t=3-3\sqrt{2}\cos t$$

There is a way to construct a possible solution using only conjugates , without applying squaring.
To get something useful, we want to multiply both side of the equation by the conjugate $3+\sqrt {15x+9}\neq 0\thinspace $ :
$$\begin{align}x\left(3+\sqrt {15x+9}\right)\left(4x^2+13x-14\right)=-15x\end{align}$$
Since $3+\sqrt {15x+9}\neq 0$ for all $x\geq -\frac 35$, this implies that $x_1=0$ is a solution. Therefore, to find other possible roots, we can proceed by dividing both side of the equation by $x\thinspace (x\neq 0\thinspace)$ :
$$
\begin{align}\left(3+\sqrt {15x+9}\right)\left(4x^2+13x-14\right)=-15\end{align}
$$
By rearranging the left-hand side of the equation, we have :
$$
\begin{align}\left(3+\sqrt {15x+9}\right)\left(4x^2+13x-12\right)-2\left(3+\sqrt {15x+9}\right)=-15\end{align}
$$
$$
\begin{align}\left(3+\sqrt {15x+9}\right)\left(4x^2+13x-12\right)=2\sqrt{15x+9}-9\end{align}
$$
Then, multiplying both side of the equation by the conjugate $2\sqrt{15x+9}+9\neq 0\thinspace$, yields :
$$
\begin{align}\left(2\sqrt{15x+9}+9\right)\left(3+\sqrt{15x+9}\right)\left(x+4\right)\left(4x-3\right)=15\left(4x-3\right)\end{align}
$$
Thus, based on the equivalence between the mathematical steps, we determine that $x_2=\frac 34$ is the second real root of the original equation, since $\thinspace 4x-3\thinspace$ is the common factor of the left and right sides of the equation.
Finally, we need to solve :
$$
\begin{align}\overbrace{\left(9+2\sqrt{15x+9}\right)}^{\ge 9}\thinspace
\overbrace {\left(3+\sqrt {15x+9}\right)}^{\ge 3}\thinspace\overbrace {\left(x+4\right)}^{>3}=15\end{align}
$$
However, we see that the last equation we obtained above has no real roots. Therefore, the original equation has only $2$ real roots : $\thinspace x\in\left\{0,\frac 34\right\}\thinspace.$
This completes the solution.

$\rm {Comment:}$
Remember that, this is not correct to generalize the method we used. The method only works on specific instances. Indeed, replace $\sqrt {15x+9}$ with the radical expression $\sqrt {7x+9}$ in the original equation, then we will definitely have to apply squaring operations and use Galois theory.
