Solve for a (complex solution)
\left\{\begin{matrix}a=-\frac{b-Pt}{tx^{2}}\text{, }&t\neq 0\text{ and }x\neq 0\\a\in \mathrm{C}\text{, }&b=Pt\text{ and }x=0\text{ and }t\neq 0\end{matrix}\right.
Solve for a
\left\{\begin{matrix}a=-\frac{b-Pt}{tx^{2}}\text{, }&t\neq 0\text{ and }x\neq 0\\a\in \mathrm{R}\text{, }&b=Pt\text{ and }x=0\text{ and }t\neq 0\end{matrix}\right.
Solve for P
P=ax^{2}+\frac{b}{t}
t\neq 0
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Pt=ax^{2}t+b
Multiply both sides of the equation by t.
ax^{2}t+b=Pt
Swap sides so that all variable terms are on the left hand side.
ax^{2}t=Pt-b
Subtract b from both sides.
tx^{2}a=Pt-b
The equation is in standard form.
\frac{tx^{2}a}{tx^{2}}=\frac{Pt-b}{tx^{2}}
Divide both sides by x^{2}t.
a=\frac{Pt-b}{tx^{2}}
Dividing by x^{2}t undoes the multiplication by x^{2}t.
Pt=ax^{2}t+b
Multiply both sides of the equation by t.
ax^{2}t+b=Pt
Swap sides so that all variable terms are on the left hand side.
ax^{2}t=Pt-b
Subtract b from both sides.
tx^{2}a=Pt-b
The equation is in standard form.
\frac{tx^{2}a}{tx^{2}}=\frac{Pt-b}{tx^{2}}
Divide both sides by x^{2}t.
a=\frac{Pt-b}{tx^{2}}
Dividing by x^{2}t undoes the multiplication by x^{2}t.
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