Solve y=sqrt{3x+1}quada+1

$$y = \sqrt { 3 x + 1 } \quad a + 1$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=-\left(3x+1\right)^{-\frac{1}{2}}\left(1-y\right)\text{, }&x\neq -\frac{1}{3}\\a\in \mathrm{C}\text{, }&y=1\text{ and }x=-\frac{1}{3}\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$\sqrt{3x+1}a+1=y$$

Subtract 1 from both sides.

$$\sqrt{3x+1}a=y-1$$

Divide both sides by \sqrt{3x+1}.

$$\frac{\sqrt{3x+1}a}{\sqrt{3x+1}}=\frac{y-1}{\sqrt{3x+1}}$$

Dividing by \sqrt{3x+1} undoes the multiplication by \sqrt{3x+1}.

$$a=\frac{y-1}{\sqrt{3x+1}}$$

Divide y-1 by \sqrt{3x+1}.

$$a=\left(3x+1\right)^{-\frac{1}{2}}\left(y-1\right)$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{1-y}{\sqrt{3x+1}}\text{, }&x>-\frac{1}{3}\\a\in \mathrm{R}\text{, }&x=-\frac{1}{3}\text{ and }y=1\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$\sqrt{3x+1}a+1=y$$

Subtract 1 from both sides.

$$\sqrt{3x+1}a=y-1$$

Divide both sides by \sqrt{3x+1}.

$$\frac{\sqrt{3x+1}a}{\sqrt{3x+1}}=\frac{y-1}{\sqrt{3x+1}}$$

Dividing by \sqrt{3x+1} undoes the multiplication by \sqrt{3x+1}.

$$a=\frac{y-1}{\sqrt{3x+1}}$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{-\left(y-1\right)^{2}+a^{2}}{3a^{2}}\text{, }&\left(y\geq 1\text{ and }a>0\right)\text{ or }\left(y\leq 1\text{ and }a<0\right)\\x\geq -\frac{1}{3}\text{, }&y=1\text{ and }a=0\end{matrix}\right.$$

Steps for Solving Radical Equations

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

$$\sqrt{3x+1}a+1=y$$

Subtract 1 from both sides.

$$\sqrt{3x+1}a=y-1$$

Divide both sides by a.

$$\frac{a\sqrt{3x+1}}{a}=\frac{y-1}{a}$$

Dividing by a undoes the multiplication by a.

$$\sqrt{3x+1}=\frac{y-1}{a}$$

Square both sides of the equation.

$$3x+1=\frac{\left(y-1\right)^{2}}{a^{2}}$$

Subtract 1 from both sides of the equation.

$$3x+1-1=\frac{\left(y-1\right)^{2}}{a^{2}}-1$$

Subtracting 1 from itself leaves 0.

$$3x=\frac{\left(y-1\right)^{2}}{a^{2}}-1$$

Subtract 1 from \frac{\left(-1+y\right)^{2}}{a^{2}}.

$$3x=\frac{\left(y-1\right)^{2}-a^{2}}{a^{2}}$$

Divide both sides by 3.

$$\frac{3x}{3}=\frac{\left(y-1\right)^{2}-a^{2}}{3a^{2}}$$

Dividing by 3 undoes the multiplication by 3.

$$x=\frac{\left(y-1\right)^{2}-a^{2}}{3a^{2}}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=-\frac{-\left(1-y\right)^{2}+a^{2}}{3a^{2}}\text{, }&\left(y=1\text{ or }arg(\frac{1-y}{a})\geq \pi \right)\text{ and }a\neq 0\\x\in \mathrm{C}\text{, }&y=1\text{ and }a=0\end{matrix}\right.$$