Solve a^3-a^2b+5ac-5bc=0

$$a ^ { 3 } - a ^ { 2 } b + 5 a c - 5 b c = 0$$

Solve for b (complex solution)

$$\left\{\begin{matrix}\\b=a\text{, }&\text{unconditionally}\\b\in \mathrm{C}\text{, }&c=-\frac{a^{2}}{5}\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract 5ac from both sides. Anything subtracted from zero gives its negation.

$$a^{3}-a^{2}b-5bc=-5ac$$

Subtract a^{3} from both sides.

$$-a^{2}b-5bc=-5ac-a^{3}$$

Combine all terms containing b.

$$\left(-a^{2}-5c\right)b=-5ac-a^{3}$$

The equation is in standard form.

$$\left(-a^{2}-5c\right)b=-a^{3}-5ac$$

Divide both sides by -a^{2}-5c.

$$\frac{\left(-a^{2}-5c\right)b}{-a^{2}-5c}=-\frac{a\left(a^{2}+5c\right)}{-a^{2}-5c}$$

Dividing by -a^{2}-5c undoes the multiplication by -a^{2}-5c.

$$b=-\frac{a\left(a^{2}+5c\right)}{-a^{2}-5c}$$

Divide -a\left(5c+a^{2}\right) by -a^{2}-5c.

$$b=a$$

Solve for b

$$\left\{\begin{matrix}\\b=a\text{, }&\text{unconditionally}\\b\in \mathrm{R}\text{, }&c=-\frac{a^{2}}{5}\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract 5ac from both sides. Anything subtracted from zero gives its negation.

$$a^{3}-a^{2}b-5bc=-5ac$$

Subtract a^{3} from both sides.

$$-a^{2}b-5bc=-5ac-a^{3}$$

Combine all terms containing b.

$$\left(-a^{2}-5c\right)b=-5ac-a^{3}$$

The equation is in standard form.

$$\left(-a^{2}-5c\right)b=-a^{3}-5ac$$

Divide both sides by -a^{2}-5c.

$$\frac{\left(-a^{2}-5c\right)b}{-a^{2}-5c}=-\frac{a\left(a^{2}+5c\right)}{-a^{2}-5c}$$

Dividing by -a^{2}-5c undoes the multiplication by -a^{2}-5c.

$$b=-\frac{a\left(a^{2}+5c\right)}{-a^{2}-5c}$$

Divide -a\left(5c+a^{2}\right) by -a^{2}-5c.

$$b=a$$

Solve for a (complex solution)

$$a=i\sqrt{5c}$$

$$a=b$$

$$a=-i\sqrt{5c}$$

Solve for a

$$\left\{\begin{matrix}\\a=b\text{, }&\text{unconditionally}\\a=\sqrt{-5c}\text{; }a=-\sqrt{-5c}\text{, }&c\leq 0\end{matrix}\right.$$