Solve (a-b)z^2-(a+b)x+2b=0

$$( a - b ) z ^ { 2 } - ( a + b ) x + 2 b = 0$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{b\left(2-z^{2}-x\right)}{z^{2}-x}\text{, }&x\neq z^{2}\\a\in \mathrm{R}\text{, }&\left(b=0\text{ and }x=z^{2}\right)\text{ or }\left(x=1\text{ and }|z|=1\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply a-b by z^{2}.

$$az^{2}-bz^{2}-\left(a+b\right)x+2b=0$$

Use the distributive property to multiply a+b by x.

$$az^{2}-bz^{2}-\left(ax+bx\right)+2b=0$$

To find the opposite of ax+bx, find the opposite of each term.

$$az^{2}-bz^{2}-ax-bx+2b=0$$

Add bz^{2} to both sides. Anything plus zero gives itself.

$$az^{2}-ax-bx+2b=bz^{2}$$

Add bx to both sides.

$$az^{2}-ax+2b=bz^{2}+bx$$

Subtract 2b from both sides.

$$az^{2}-ax=bz^{2}+bx-2b$$

Combine all terms containing a.

$$\left(z^{2}-x\right)a=bz^{2}+bx-2b$$

The equation is in standard form.

$$\left(z^{2}-x\right)a=bx+bz^{2}-2b$$

Divide both sides by -x+z^{2}.

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

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

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

Solve for b

$$\left\{\begin{matrix}b=-\frac{a\left(z^{2}-x\right)}{2-z^{2}-x}\text{, }&x\neq 2-z^{2}\\b\in \mathrm{R}\text{, }&\left(a=0\text{ and }x=2-z^{2}\right)\text{ or }\left(x=1\text{ and }|z|=1\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply a-b by z^{2}.

$$az^{2}-bz^{2}-\left(a+b\right)x+2b=0$$

Use the distributive property to multiply a+b by x.

$$az^{2}-bz^{2}-\left(ax+bx\right)+2b=0$$

To find the opposite of ax+bx, find the opposite of each term.

$$az^{2}-bz^{2}-ax-bx+2b=0$$

Subtract az^{2} from both sides. Anything subtracted from zero gives its negation.

$$-bz^{2}-ax-bx+2b=-az^{2}$$

Add ax to both sides.

$$-bz^{2}-bx+2b=-az^{2}+ax$$

Reorder the terms.

$$-bx-bz^{2}+2b=ax-az^{2}$$

Combine all terms containing b.

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

The equation is in standard form.

$$\left(2-z^{2}-x\right)b=ax-az^{2}$$

Divide both sides by -x-z^{2}+2.

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

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

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