Square of sum: Data with powers and roots

Solve for X:

$x+\sqrt{x}=-\sqrt{x}$

$0$

$\frac{\sqrt{2x^2+4xy+2y^2+(x+y)^2}}{(x+y)}=$

$\sqrt{3}$

Simply the following expression:

$(x+\sqrt{x})^2$

$x\lbrack x+2\sqrt{x}+1\rbrack$

Solve the following equation:

$\sqrt{x+1}\times\sqrt{x+2}=x+3$

$x=-\frac{7}{3}$

Look at the following equation:

$\frac{\sqrt{x}-\sqrt{x+1}}{x+1}=1$

This can also be written as:

$x[A(x+B)-x^3]=0$

Calculate A and B.

B=1 , A=4

Solve the following system of equations:

$\begin{cases} \sqrt{x}+\sqrt{y}=\sqrt{\sqrt{61}+6} \\ xy=9 \end{cases}$

$x=\frac{\sqrt{61}}{2}-2.5$

$y=\frac{\sqrt{61}}{2}+2.5$

or

$x=\frac{\sqrt{61}}{2}+2.5$

$y=\frac{\sqrt{61}}{2}-2.5$

Look at the following equation:

$\frac{\sqrt{x}+\sqrt{x+1}}{x+1}=1$

The same equation can be presented as follows:

$x[A(x+B)-x^3]=0$

Calculate A and B.

B=1 , A=4