Solve x/y=sqrt{x^2-a+6}

$$x / y = \sqrt { x ^ { 2 } - a + 6 }$$

Solve for a

$$a=x^{2}-\left(\frac{x}{y}\right)^{2}+6$$

$$\left(x\geq 0\text{ and }y>0\right)\text{ or }\left(x\leq 0\text{ and }y<0\right)$$

Steps for Solving Radical Equations

Multiply both sides of the equation by y.

$$x=y\sqrt{x^{2}-a+6}$$

Swap sides so that all variable terms are on the left hand side.

$$y\sqrt{x^{2}-a+6}=x$$

Divide both sides by y.

$$\frac{y\sqrt{-a+x^{2}+6}}{y}=\frac{x}{y}$$

Dividing by y undoes the multiplication by y.

$$\sqrt{-a+x^{2}+6}=\frac{x}{y}$$

Square both sides of the equation.

$$-a+x^{2}+6=\frac{x^{2}}{y^{2}}$$

Subtract x^{2}+6 from both sides of the equation.

$$-a+x^{2}+6-\left(x^{2}+6\right)=\frac{x^{2}}{y^{2}}-\left(x^{2}+6\right)$$

Subtracting x^{2}+6 from itself leaves 0.

$$-a=\frac{x^{2}}{y^{2}}-\left(x^{2}+6\right)$$

Subtract x^{2}+6 from \frac{x^{2}}{y^{2}}.

$$-a=-x^{2}+\frac{x^{2}}{y^{2}}-6$$

Divide both sides by -1.

$$\frac{-a}{-1}=\frac{-x^{2}+\frac{x^{2}}{y^{2}}-6}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$a=\frac{-x^{2}+\frac{x^{2}}{y^{2}}-6}{-1}$$

Divide -x^{2}-6+\frac{x^{2}}{y^{2}} by -1.

$$a=x^{2}-\frac{x^{2}}{y^{2}}+6$$

Solve for a (complex solution)

$$a=x^{2}-\left(\frac{x}{y}\right)^{2}+6$$

$$y\neq 0\text{ and }\left(|arg(\sqrt{\left(\frac{x}{y}\right)^{2}}y)-arg(x)|<\pi \text{ or }x=0\right)$$