Solve y=-ax^2-bx-c

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

Solve for a (complex solution)

$$\left\{\begin{matrix}a=-\frac{bx+y+c}{x^{2}}\text{, }&x\neq 0\\a\in \mathrm{C}\text{, }&y=-c\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$\left(-a\right)x^{2}-bx-c=y$$

Add bx to both sides.

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

Add c to both sides.

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

Reorder the terms.

$$-ax^{2}=bx+y+c$$

The equation is in standard form.

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

Divide both sides by -x^{2}.

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

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

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

Divide y+c+bx by -x^{2}.

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

Solve for b (complex solution)

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

Steps for Solving Linear Equation

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

$$\left(-a\right)x^{2}-bx-c=y$$

Subtract \left(-a\right)x^{2} from both sides.

$$-bx-c=y-\left(-a\right)x^{2}$$

Add c to both sides.

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

Multiply -1 and -1 to get 1.

$$-bx=y+ax^{2}+c$$

The equation is in standard form.

$$\left(-x\right)b=ax^{2}+y+c$$

Divide both sides by -x.

$$\frac{\left(-x\right)b}{-x}=\frac{ax^{2}+y+c}{-x}$$

Dividing by -x undoes the multiplication by -x.

$$b=\frac{ax^{2}+y+c}{-x}$$

Divide y+ax^{2}+c by -x.

$$b=-\frac{ax^{2}+y+c}{x}$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{bx+y+c}{x^{2}}\text{, }&x\neq 0\\a\in \mathrm{R}\text{, }&y=-c\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$\left(-a\right)x^{2}-bx-c=y$$

Add bx to both sides.

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

Add c to both sides.

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

Reorder the terms.

$$-ax^{2}=bx+y+c$$

The equation is in standard form.

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

Divide both sides by -x^{2}.

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

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

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

Divide y+bx+c by -x^{2}.

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

Solve for b

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

Steps for Solving Linear Equation

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

$$\left(-a\right)x^{2}-bx-c=y$$

Subtract \left(-a\right)x^{2} from both sides.

$$-bx-c=y-\left(-a\right)x^{2}$$

Add c to both sides.

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

Multiply -1 and -1 to get 1.

$$-bx=y+ax^{2}+c$$

The equation is in standard form.

$$\left(-x\right)b=ax^{2}+y+c$$

Divide both sides by -x.

$$\frac{\left(-x\right)b}{-x}=\frac{ax^{2}+y+c}{-x}$$

Dividing by -x undoes the multiplication by -x.

$$b=\frac{ax^{2}+y+c}{-x}$$

Divide y+ax^{2}+c by -x.

$$b=-\frac{ax^{2}+y+c}{x}$$