Solve 1/2x^2+bx+c=0

$$\frac { 1 } { 2 } x ^ { 2 } + b x + c = 0$$

Solve for b (complex solution)

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

Steps for Solving Linear Equation

Subtract \frac{1}{2}x^{2} from both sides. Anything subtracted from zero gives its negation.

$$bx+c=-\frac{1}{2}x^{2}$$

Subtract c from both sides.

$$bx=-\frac{1}{2}x^{2}-c$$

The equation is in standard form.

$$xb=-\frac{x^{2}}{2}-c$$

Divide both sides by x.

$$\frac{xb}{x}=\frac{-\frac{x^{2}}{2}-c}{x}$$

Dividing by x undoes the multiplication by x.

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

Divide -\frac{x^{2}}{2}-c by x.

$$b=-\frac{x}{2}-\frac{c}{x}$$

Solve for b

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

Steps for Solving Linear Equation

Subtract \frac{1}{2}x^{2} from both sides. Anything subtracted from zero gives its negation.

$$bx+c=-\frac{1}{2}x^{2}$$

Subtract c from both sides.

$$bx=-\frac{1}{2}x^{2}-c$$

The equation is in standard form.

$$xb=-\frac{x^{2}}{2}-c$$

Divide both sides by x.

$$\frac{xb}{x}=\frac{-\frac{x^{2}}{2}-c}{x}$$

Dividing by x undoes the multiplication by x.

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

Divide -\frac{x^{2}}{2}-c by x.

$$b=-\frac{x}{2}-\frac{c}{x}$$

Solve for c

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

Solution Steps

Subtract \frac{1}{2}x^{2} from both sides. Anything subtracted from zero gives its negation.

$$bx+c=-\frac{1}{2}x^{2}$$

Subtract bx from both sides.

$$c=-\frac{1}{2}x^{2}-bx$$