Solve f(p)^2=12p^2-36p+31

$$f ( p ) ^ { 2 } = 12 p ^ { 2 } - 36 p + 31$$

Solve for f

$$f=12-\frac{36}{p}+\frac{31}{p^{2}}$$

$$p\neq 0$$

Steps for Solving Linear Equation

The equation is in standard form.

$$p^{2}f=12p^{2}-36p+31$$

Divide both sides by p^{2}.

$$\frac{p^{2}f}{p^{2}}=\frac{12p^{2}-36p+31}{p^{2}}$$

Dividing by p^{2} undoes the multiplication by p^{2}.

$$f=\frac{12p^{2}-36p+31}{p^{2}}$$

Divide 12p^{2}-36p+31 by p^{2}.

$$f=12+\frac{31-36p}{p^{2}}$$

Solve for p (complex solution)

$$\left\{\begin{matrix}p=\frac{\sqrt{31f-48}-18}{f-12}\text{; }p=-\frac{\sqrt{31f-48}+18}{f-12}\text{, }&f\neq 12\\p=\frac{31}{36}\text{, }&f=12\end{matrix}\right.$$

Solve for p

$$\left\{\begin{matrix}p=\frac{\sqrt{31f-48}-18}{f-12}\text{; }p=-\frac{\sqrt{31f-48}+18}{f-12}\text{, }&f\neq 12\text{ and }f\geq \frac{48}{31}\\p=\frac{31}{36}\text{, }&f=12\end{matrix}\right.$$