Solve ax^2+24x+b=m^2x^2-bmx+9

$$a x ^ { 2 } + 24 x + b = m ^ { 2 } x ^ { 2 } - b m x + 9$$

Solve for b (complex solution)

$$\left\{\begin{matrix}b=-\frac{ax^{2}+24x-\left(mx\right)^{2}-9}{mx+1}\text{, }&m=0\text{ or }x\neq -\frac{1}{m}\\b\in \mathrm{C}\text{, }&a=10m^{2}+24m\text{ and }x=-\frac{1}{m}\text{ and }m\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract m^{2}x^{2}-bmx from both sides.

$$ax^{2}+24x+b-\left(m^{2}x^{2}-bmx\right)=9$$

To find the opposite of m^{2}x^{2}-bmx, find the opposite of each term.

$$ax^{2}+24x+b-m^{2}x^{2}+bmx=9$$

Subtract ax^{2} from both sides.

$$24x+b-m^{2}x^{2}+bmx=9-ax^{2}$$

Subtract 24x from both sides.

$$b-m^{2}x^{2}+bmx=9-ax^{2}-24x$$

Add m^{2}x^{2} to both sides.

$$b+bmx=9-ax^{2}-24x+m^{2}x^{2}$$

Combine all terms containing b.

$$\left(1+mx\right)b=9-ax^{2}-24x+m^{2}x^{2}$$

The equation is in standard form.

$$\left(mx+1\right)b=m^{2}x^{2}-ax^{2}-24x+9$$

Divide both sides by mx+1.

$$\frac{\left(mx+1\right)b}{mx+1}=\frac{m^{2}x^{2}-ax^{2}-24x+9}{mx+1}$$

Dividing by mx+1 undoes the multiplication by mx+1.

$$b=\frac{m^{2}x^{2}-ax^{2}-24x+9}{mx+1}$$

Solve for b

$$\left\{\begin{matrix}b=-\frac{ax^{2}+24x-\left(mx\right)^{2}-9}{mx+1}\text{, }&m=0\text{ or }x\neq -\frac{1}{m}\\b\in \mathrm{R}\text{, }&a=10m^{2}+24m\text{ and }x=-\frac{1}{m}\text{ and }m\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract m^{2}x^{2}-bmx from both sides.

$$ax^{2}+24x+b-\left(m^{2}x^{2}-bmx\right)=9$$

To find the opposite of m^{2}x^{2}-bmx, find the opposite of each term.

$$ax^{2}+24x+b-m^{2}x^{2}+bmx=9$$

Subtract ax^{2} from both sides.

$$24x+b-m^{2}x^{2}+bmx=9-ax^{2}$$

Subtract 24x from both sides.

$$b-m^{2}x^{2}+bmx=9-ax^{2}-24x$$

Add m^{2}x^{2} to both sides.

$$b+bmx=9-ax^{2}-24x+m^{2}x^{2}$$

Combine all terms containing b.

$$\left(1+mx\right)b=9-ax^{2}-24x+m^{2}x^{2}$$

The equation is in standard form.

$$\left(mx+1\right)b=m^{2}x^{2}-ax^{2}-24x+9$$

Divide both sides by mx+1.

$$\frac{\left(mx+1\right)b}{mx+1}=\frac{m^{2}x^{2}-ax^{2}-24x+9}{mx+1}$$

Dividing by mx+1 undoes the multiplication by mx+1.

$$b=\frac{m^{2}x^{2}-ax^{2}-24x+9}{mx+1}$$