Solve left(36-2bright)left(1-mright)^2=5

$$\left( 36-2b \right) { \left(1-m \right) }^{ 2 } = 5$$

Solve for b

$$b=\frac{36m^{2}-72m+31}{2\left(m-1\right)^{2}}$$

$$m\neq 1$$

Steps for Solving Linear Equation

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1-m\right)^{2}.

$$\left(36-2b\right)\left(1-2m+m^{2}\right)=5$$

Use the distributive property to multiply 36-2b by 1-2m+m^{2}.

$$36-72m+36m^{2}-2b+4bm-2bm^{2}=5$$

Subtract 36 from both sides.

$$-72m+36m^{2}-2b+4bm-2bm^{2}=5-36$$

Subtract 36 from 5 to get -31.

$$-72m+36m^{2}-2b+4bm-2bm^{2}=-31$$

Add 72m to both sides.

$$36m^{2}-2b+4bm-2bm^{2}=-31+72m$$

Subtract 36m^{2} from both sides.

$$-2b+4bm-2bm^{2}=-31+72m-36m^{2}$$

Combine all terms containing b.

$$\left(-2+4m-2m^{2}\right)b=-31+72m-36m^{2}$$

The equation is in standard form.

$$\left(-2m^{2}+4m-2\right)b=-36m^{2}+72m-31$$

Divide both sides by -2m^{2}+4m-2.

$$\frac{\left(-2m^{2}+4m-2\right)b}{-2m^{2}+4m-2}=\frac{-36m^{2}+72m-31}{-2m^{2}+4m-2}$$

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

$$b=\frac{-36m^{2}+72m-31}{-2m^{2}+4m-2}$$

Divide -31+72m-36m^{2} by -2m^{2}+4m-2.

$$b=-\frac{-36m^{2}+72m-31}{2\left(m-1\right)^{2}}$$

Solve for m

$$m=\frac{\sqrt{\frac{10}{18-b}}\left(\sqrt{10\left(18-b\right)}-5\right)}{10}$$

$$m=\frac{\sqrt{\frac{10}{18-b}}\left(\sqrt{10\left(18-b\right)}+5\right)}{10}\text{, }b<18$$