Solve operatornametax^2-bx+c=0

$$\operatorname { tax } ^ { 2 } - b x + c = 0$$

Solve for b (complex solution)

$$\left\{\begin{matrix}b=\frac{\left(atx\right)^{2}+c}{x}\text{, }&x\neq 0\\b\in \mathrm{C}\text{, }&c=0\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Expand \left(tax\right)^{2}.

$$t^{2}a^{2}x^{2}-bx+c=0$$

Subtract t^{2}a^{2}x^{2} from both sides. Anything subtracted from zero gives its negation.

$$-bx+c=-t^{2}a^{2}x^{2}$$

Subtract c from both sides.

$$-bx=-t^{2}a^{2}x^{2}-c$$

Reorder the terms.

$$-bx=-a^{2}t^{2}x^{2}-c$$

The equation is in standard form.

$$\left(-x\right)b=-a^{2}t^{2}x^{2}-c$$

Divide both sides by -x.

$$\frac{\left(-x\right)b}{-x}=\frac{-a^{2}t^{2}x^{2}-c}{-x}$$

Dividing by -x undoes the multiplication by -x.

$$b=\frac{-a^{2}t^{2}x^{2}-c}{-x}$$

Divide -c-x^{2}t^{2}a^{2} by -x.

$$b=xa^{2}t^{2}+\frac{c}{x}$$

Solve for b

$$\left\{\begin{matrix}b=\frac{\left(atx\right)^{2}+c}{x}\text{, }&x\neq 0\\b\in \mathrm{R}\text{, }&c=0\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Expand \left(tax\right)^{2}.

$$t^{2}a^{2}x^{2}-bx+c=0$$

Subtract t^{2}a^{2}x^{2} from both sides. Anything subtracted from zero gives its negation.

$$-bx+c=-t^{2}a^{2}x^{2}$$

Subtract c from both sides.

$$-bx=-t^{2}a^{2}x^{2}-c$$

Reorder the terms.

$$-bx=-a^{2}t^{2}x^{2}-c$$

The equation is in standard form.

$$\left(-x\right)b=-a^{2}t^{2}x^{2}-c$$

Divide both sides by -x.

$$\frac{\left(-x\right)b}{-x}=\frac{-a^{2}t^{2}x^{2}-c}{-x}$$

Dividing by -x undoes the multiplication by -x.

$$b=\frac{-a^{2}t^{2}x^{2}-c}{-x}$$

Divide -a^{2}t^{2}x^{2}-c by -x.

$$b=xa^{2}t^{2}+\frac{c}{x}$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a\in \mathrm{C}\text{, }&\left(c=bx\text{ and }t=0\right)\text{ or }\left(c=0\text{ and }x=0\right)\\a=-\frac{\sqrt{bx-c}}{tx}\text{; }a=\frac{\sqrt{bx-c}}{tx}\text{, }&x\neq 0\text{ and }t\neq 0\end{matrix}\right.$$

Solve for a

$$\left\{\begin{matrix}a\in \mathrm{R}\text{, }&\left(c=bx\text{ and }t=0\right)\text{ or }\left(c=0\text{ and }x=0\right)\\a=-\frac{\sqrt{bx-c}}{tx}\text{; }a=\frac{\sqrt{bx-c}}{tx}\text{, }&x\neq 0\text{ and }t\neq 0\text{ and }c\leq bx\end{matrix}\right.$$