Solve ax^2-5bx+4a=0

$$a x ^ { 2 } - 5 b x + 4 a = 0$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=\frac{5bx}{x^{2}+4}\text{, }&x\neq -2i\text{ and }x\neq 2i\\a\in \mathrm{C}\text{, }&\left(x=2i\text{ or }x=-2i\right)\text{ and }b=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Add 5bx to both sides. Anything plus zero gives itself.

$$ax^{2}+4a=5bx$$

Combine all terms containing a.

$$\left(x^{2}+4\right)a=5bx$$

Divide both sides by x^{2}+4.

$$\frac{\left(x^{2}+4\right)a}{x^{2}+4}=\frac{5bx}{x^{2}+4}$$

Dividing by x^{2}+4 undoes the multiplication by x^{2}+4.

$$a=\frac{5bx}{x^{2}+4}$$

Solve for b (complex solution)

$$\left\{\begin{matrix}b=\frac{a\left(x^{2}+4\right)}{5x}\text{, }&x\neq 0\\b\in \mathrm{C}\text{, }&a=0\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$-5bx+4a=-ax^{2}$$

Subtract 4a from both sides.

$$-5bx=-ax^{2}-4a$$

The equation is in standard form.

$$\left(-5x\right)b=-ax^{2}-4a$$

Divide both sides by -5x.

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

Dividing by -5x undoes the multiplication by -5x.

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

Divide a\left(x+2i\right)\left(-x+2i\right) by -5x.

$$b=\frac{ax}{5}+\frac{4a}{5x}$$

Solve for a

$$a=\frac{5bx}{x^{2}+4}$$

Steps for Solving Linear Equation

Add 5bx to both sides. Anything plus zero gives itself.

$$ax^{2}+4a=5bx$$

Combine all terms containing a.

$$\left(x^{2}+4\right)a=5bx$$

Divide both sides by x^{2}+4.

$$\frac{\left(x^{2}+4\right)a}{x^{2}+4}=\frac{5bx}{x^{2}+4}$$

Dividing by x^{2}+4 undoes the multiplication by x^{2}+4.

$$a=\frac{5bx}{x^{2}+4}$$

Solve for b

$$\left\{\begin{matrix}b=\frac{a\left(x^{2}+4\right)}{5x}\text{, }&x\neq 0\\b\in \mathrm{R}\text{, }&a=0\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$-5bx+4a=-ax^{2}$$

Subtract 4a from both sides.

$$-5bx=-ax^{2}-4a$$

The equation is in standard form.

$$\left(-5x\right)b=-ax^{2}-4a$$

Divide both sides by -5x.

$$\frac{\left(-5x\right)b}{-5x}=-\frac{a\left(x^{2}+4\right)}{-5x}$$

Dividing by -5x undoes the multiplication by -5x.

$$b=-\frac{a\left(x^{2}+4\right)}{-5x}$$

Divide -a\left(x^{2}+4\right) by -5x.

$$b=\frac{ax}{5}+\frac{4a}{5x}$$