Solve ax^2-(ab+1)x+b=0

$$a x ^ { 2 } - ( a b + 1 ) x + b = 0$$

Solve for a (complex solution)

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

Steps for Solving Linear Equation

Use the distributive property to multiply ab+1 by x.

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

To find the opposite of abx+x, find the opposite of each term.

$$ax^{2}-abx-x+b=0$$

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

$$ax^{2}-abx+b=x$$

Subtract b from both sides.

$$ax^{2}-abx=x-b$$

Combine all terms containing a.

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

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

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

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

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

Divide x-b by x^{2}-bx.

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

Solve for b (complex solution)

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

Steps for Solving Linear Equation

Use the distributive property to multiply ab+1 by x.

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

To find the opposite of abx+x, find the opposite of each term.

$$ax^{2}-abx-x+b=0$$

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

$$-abx-x+b=-ax^{2}$$

Add x to both sides.

$$-abx+b=-ax^{2}+x$$

Combine all terms containing b.

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

The equation is in standard form.

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

Divide both sides by 1-ax.

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

Dividing by 1-ax undoes the multiplication by 1-ax.

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

Divide x\left(-ax+1\right) by 1-ax.

$$b=x$$

Solve for a

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

Steps for Solving Linear Equation

Use the distributive property to multiply ab+1 by x.

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

To find the opposite of abx+x, find the opposite of each term.

$$ax^{2}-abx-x+b=0$$

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

$$ax^{2}-abx+b=x$$

Subtract b from both sides.

$$ax^{2}-abx=x-b$$

Combine all terms containing a.

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

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

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

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

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

Divide x-b by x^{2}-bx.

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

Solve for b

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

Steps for Solving Linear Equation

Use the distributive property to multiply ab+1 by x.

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

To find the opposite of abx+x, find the opposite of each term.

$$ax^{2}-abx-x+b=0$$

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

$$-abx-x+b=-ax^{2}$$

Add x to both sides.

$$-abx+b=-ax^{2}+x$$

Combine all terms containing b.

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

The equation is in standard form.

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

Divide both sides by 1-ax.

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

Dividing by 1-ax undoes the multiplication by 1-ax.

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

Divide x\left(-ax+1\right) by 1-ax.

$$b=x$$