Solve 82=a(s+4)^2+1

$$82 = a ( s + 4 ) ^ { 2 } + 1$$

Solve for a

$$a=\frac{81}{\left(s+4\right)^{2}}$$

$$s\neq -4$$

Steps for Solving Linear Equation

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(s+4\right)^{2}.

$$82=a\left(s^{2}+8s+16\right)+1$$

Use the distributive property to multiply a by s^{2}+8s+16.

$$82=as^{2}+8as+16a+1$$

Swap sides so that all variable terms are on the left hand side.

$$as^{2}+8as+16a+1=82$$

Subtract 1 from both sides.

$$as^{2}+8as+16a=82-1$$

Subtract 1 from 82 to get 81.

$$as^{2}+8as+16a=81$$

Combine all terms containing a.

$$\left(s^{2}+8s+16\right)a=81$$

Divide both sides by s^{2}+8s+16.

$$\frac{\left(s^{2}+8s+16\right)a}{s^{2}+8s+16}=\frac{81}{s^{2}+8s+16}$$

Dividing by s^{2}+8s+16 undoes the multiplication by s^{2}+8s+16.

$$a=\frac{81}{s^{2}+8s+16}$$

Divide 81 by s^{2}+8s+16.

$$a=\frac{81}{\left(s+4\right)^{2}}$$

Solve for s (complex solution)

$$s=9a^{-\frac{1}{2}}-4$$

$$s=-9a^{-\frac{1}{2}}-4\text{, }a\neq 0$$

Solve for s

$$s=-4+\frac{9}{\sqrt{a}}$$

$$s=-4-\frac{9}{\sqrt{a}}\text{, }a>0$$