Solve y=3x/4+sqrt{ax^2}

$$y = \frac { 3 x } { 4 } + \sqrt { a x ^ { 2 } }$$

Solve for a

$$\left\{\begin{matrix}a=\frac{\left(\frac{4y}{x}-3\right)^{2}}{16}\text{, }&x\neq 0\text{ and }-\frac{3x}{4}+y\geq 0\\a\in \mathrm{R}\text{, }&y=0\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Radical Equations

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

$$\frac{3x}{4}+\sqrt{ax^{2}}=y$$

Subtract \frac{3x}{4} from both sides.

$$\sqrt{ax^{2}}=y-\frac{3x}{4}$$

Multiply both sides of the equation by 4.

$$4\sqrt{ax^{2}}=4y-3x$$

Divide both sides by 4.

$$\frac{4\sqrt{x^{2}a}}{4}=\frac{4y-3x}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$\sqrt{x^{2}a}=\frac{4y-3x}{4}$$

Divide 4y-3x by 4.

$$\sqrt{x^{2}a}=-\frac{3x}{4}+y$$

Square both sides of the equation.

$$x^{2}a=\frac{\left(4y-3x\right)^{2}}{16}$$

Divide both sides by x^{2}.

$$\frac{x^{2}a}{x^{2}}=\frac{\left(4y-3x\right)^{2}}{16x^{2}}$$

Dividing by x^{2} undoes the multiplication by x^{2}.

$$a=\frac{\left(4y-3x\right)^{2}}{16x^{2}}$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=\frac{\left(\frac{4y}{x}-3\right)^{2}}{16}\text{, }&x\neq 0\text{ and }\left(y=\frac{3x}{4}\text{ or }arg(-\frac{3x}{4}+y)<\pi \right)\\a\in \mathrm{C}\text{, }&y=0\text{ and }x=0\end{matrix}\right.$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{4\left(4\sqrt{ay^{2}}-3y\right)}{16a-9}\text{; }x=-\frac{4\left(4\sqrt{ay^{2}}+3y\right)}{16a-9}\text{, }&a\neq \frac{9}{16}\text{ and }arg(-\frac{3x}{4}+y)<\pi \\x=\frac{2y}{3}\text{, }&a=\frac{9}{16}\text{ and }y\neq 0\text{ and }arg(-\frac{3x}{4}+y)<\pi \\x\in \mathrm{C}\text{, }&a=\frac{9}{16}\text{ and }y=0\text{ and }arg(x)\geq \pi \\x=\frac{4y}{3}\text{, }&y=0\text{ or }a=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{4y}{4\sqrt{a}-3}\text{, }&\left(a<\frac{9}{16}\text{ or }y\geq 0\right)\text{ and }\left(a=0\text{ or }a>\frac{9}{16}\text{ or }y\leq 0\right)\text{ and }a\geq 0\text{ and }a\neq \frac{9}{16}\\x=\frac{4y}{4\sqrt{a}+3}\text{, }&a=0\text{ or }\left(y\geq 0\text{ and }a\geq 0\text{ and }a\neq \frac{9}{16}\right)\\x=\frac{2y}{3}\text{, }&y=0\text{ or }\left(a=\frac{9}{16}\text{ and }y>0\right)\\x\leq 0\text{, }&a=\frac{9}{16}\text{ and }y=0\end{matrix}\right.$$