Solve abx^2-left(a+bright)x+1=0

$$ab { x }^{ 2 } - \left( a+b \right) x+1 = 0$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=\frac{1}{x}\text{, }&x\neq 0\\a\in \mathrm{C}\text{, }&b=\frac{1}{x}\text{ and }x\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$abx^{2}-\left(ax+bx\right)+1=0$$

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

$$abx^{2}-ax-bx+1=0$$

Add bx to both sides. Anything plus zero gives itself.

$$abx^{2}-ax+1=bx$$

Subtract 1 from both sides.

$$abx^{2}-ax=bx-1$$

Combine all terms containing a.

$$\left(bx^{2}-x\right)a=bx-1$$

Divide both sides by bx^{2}-x.

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

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

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

Divide -1+bx by bx^{2}-x.

$$a=\frac{1}{x}$$

Solve for b (complex solution)

$$\left\{\begin{matrix}b=\frac{1}{x}\text{, }&x\neq 0\\b\in \mathrm{C}\text{, }&a=\frac{1}{x}\text{ and }x\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$abx^{2}-\left(ax+bx\right)+1=0$$

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

$$abx^{2}-ax-bx+1=0$$

Add ax to both sides. Anything plus zero gives itself.

$$abx^{2}-bx+1=ax$$

Subtract 1 from both sides.

$$abx^{2}-bx=ax-1$$

Combine all terms containing b.

$$\left(ax^{2}-x\right)b=ax-1$$

Divide both sides by ax^{2}-x.

$$\frac{\left(ax^{2}-x\right)b}{ax^{2}-x}=\frac{ax-1}{ax^{2}-x}$$

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

$$b=\frac{ax-1}{ax^{2}-x}$$

Divide -1+ax by ax^{2}-x.

$$b=\frac{1}{x}$$

Solve for a

$$\left\{\begin{matrix}a=\frac{1}{x}\text{, }&x\neq 0\\a\in \mathrm{R}\text{, }&b=\frac{1}{x}\text{ and }x\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$abx^{2}-\left(ax+bx\right)+1=0$$

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

$$abx^{2}-ax-bx+1=0$$

Add bx to both sides. Anything plus zero gives itself.

$$abx^{2}-ax+1=bx$$

Subtract 1 from both sides.

$$abx^{2}-ax=bx-1$$

Combine all terms containing a.

$$\left(bx^{2}-x\right)a=bx-1$$

Divide both sides by bx^{2}-x.

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

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

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

Divide -1+bx by bx^{2}-x.

$$a=\frac{1}{x}$$

Solve for b

$$\left\{\begin{matrix}b=\frac{1}{x}\text{, }&x\neq 0\\b\in \mathrm{R}\text{, }&a=\frac{1}{x}\text{ and }x\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$abx^{2}-\left(ax+bx\right)+1=0$$

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

$$abx^{2}-ax-bx+1=0$$

Add ax to both sides. Anything plus zero gives itself.

$$abx^{2}-bx+1=ax$$

Subtract 1 from both sides.

$$abx^{2}-bx=ax-1$$

Combine all terms containing b.

$$\left(ax^{2}-x\right)b=ax-1$$

Divide both sides by ax^{2}-x.

$$\frac{\left(ax^{2}-x\right)b}{ax^{2}-x}=\frac{ax-1}{ax^{2}-x}$$

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

$$b=\frac{ax-1}{ax^{2}-x}$$

Divide -1+ax by ax^{2}-x.

$$b=\frac{1}{x}$$