Solve y=1/2x^2-ax+a+3

$$y = \frac{ 1 }{ 2 } { x }^{ 2 } -ax+a+3$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=-\frac{x^{2}-2y+6}{2\left(1-x\right)}\text{, }&x\neq 1\\a\in \mathrm{C}\text{, }&y=\frac{7}{2}\text{ and }x=1\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$\frac{1}{2}x^{2}-ax+a+3=y$$

Subtract \frac{1}{2}x^{2} from both sides.

$$-ax+a+3=y-\frac{1}{2}x^{2}$$

Subtract 3 from both sides.

$$-ax+a=y-\frac{1}{2}x^{2}-3$$

Combine all terms containing a.

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

The equation is in standard form.

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

Divide both sides by -x+1.

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

Dividing by -x+1 undoes the multiplication by -x+1.

$$a=\frac{-\frac{x^{2}}{2}+y-3}{1-x}$$

Divide y-\frac{x^{2}}{2}-3 by -x+1.

$$a=\frac{-x^{2}+2y-6}{2\left(1-x\right)}$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{x^{2}-2y+6}{2\left(1-x\right)}\text{, }&x\neq 1\\a\in \mathrm{R}\text{, }&y=\frac{7}{2}\text{ and }x=1\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$\frac{1}{2}x^{2}-ax+a+3=y$$

Subtract \frac{1}{2}x^{2} from both sides.

$$-ax+a+3=y-\frac{1}{2}x^{2}$$

Subtract 3 from both sides.

$$-ax+a=y-\frac{1}{2}x^{2}-3$$

Combine all terms containing a.

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

The equation is in standard form.

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

Divide both sides by -x+1.

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

Dividing by -x+1 undoes the multiplication by -x+1.

$$a=\frac{-\frac{x^{2}}{2}+y-3}{1-x}$$

Divide y-\frac{x^{2}}{2}-3 by -x+1.

$$a=\frac{-x^{2}+2y-6}{2\left(1-x\right)}$$

Solve for x (complex solution)

$$x=\sqrt{2y+a^{2}-2a-6}+a$$

$$x=-\sqrt{2y+a^{2}-2a-6}+a$$

Solve for x

$$x=\sqrt{2y+a^{2}-2a-6}+a$$

$$x=-\sqrt{2y+a^{2}-2a-6}+a\text{, }y\geq -\frac{a^{2}}{2}+a+3$$