Solve y=aleft(x-pright)^2+q

$$y = a { \left(x-p \right) }^{ 2 } +q$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=-\frac{q-y}{\left(x-p\right)^{2}}\text{, }&x\neq p\\a\in \mathrm{C}\text{, }&y=q\text{ and }x=p\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$y=a\left(x^{2}-2xp+p^{2}\right)+q$$

Use the distributive property to multiply a by x^{2}-2xp+p^{2}.

$$y=ax^{2}-2axp+ap^{2}+q$$

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

$$ax^{2}-2axp+ap^{2}+q=y$$

Subtract q from both sides.

$$ax^{2}-2axp+ap^{2}=y-q$$

Combine all terms containing a.

$$\left(x^{2}-2xp+p^{2}\right)a=y-q$$

The equation is in standard form.

$$\left(x^{2}-2px+p^{2}\right)a=y-q$$

Divide both sides by x^{2}-2xp+p^{2}.

$$\frac{\left(x^{2}-2px+p^{2}\right)a}{x^{2}-2px+p^{2}}=\frac{y-q}{x^{2}-2px+p^{2}}$$

Dividing by x^{2}-2xp+p^{2} undoes the multiplication by x^{2}-2xp+p^{2}.

$$a=\frac{y-q}{x^{2}-2px+p^{2}}$$

Divide y-q by x^{2}-2xp+p^{2}.

$$a=\frac{y-q}{\left(x-p\right)^{2}}$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{q-y}{\left(x-p\right)^{2}}\text{, }&x\neq p\\a\in \mathrm{R}\text{, }&y=q\text{ and }x=p\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$y=a\left(x^{2}-2xp+p^{2}\right)+q$$

Use the distributive property to multiply a by x^{2}-2xp+p^{2}.

$$y=ax^{2}-2axp+ap^{2}+q$$

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

$$ax^{2}-2axp+ap^{2}+q=y$$

Subtract q from both sides.

$$ax^{2}-2axp+ap^{2}=y-q$$

Combine all terms containing a.

$$\left(x^{2}-2xp+p^{2}\right)a=y-q$$

The equation is in standard form.

$$\left(x^{2}-2px+p^{2}\right)a=y-q$$

Divide both sides by x^{2}-2xp+p^{2}.

$$\frac{\left(x^{2}-2px+p^{2}\right)a}{x^{2}-2px+p^{2}}=\frac{y-q}{x^{2}-2px+p^{2}}$$

Dividing by x^{2}-2xp+p^{2} undoes the multiplication by x^{2}-2xp+p^{2}.

$$a=\frac{y-q}{x^{2}-2px+p^{2}}$$

Divide y-q by x^{2}-2xp+p^{2}.

$$a=\frac{y-q}{\left(x-p\right)^{2}}$$

Solve for p (complex solution)

$$\left\{\begin{matrix}p=x-ia^{-\frac{1}{2}}\sqrt{q-y}\text{; }p=x+ia^{-\frac{1}{2}}\sqrt{q-y}\text{, }&a\neq 0\\p\in \mathrm{C}\text{, }&y=q\text{ and }a=0\end{matrix}\right.$$

Solve for p

$$\left\{\begin{matrix}p=\frac{\sqrt{a}x+\sqrt{y-q}}{\sqrt{a}}\text{; }p=\frac{\sqrt{a}x-\sqrt{y-q}}{\sqrt{a}}\text{, }&y\geq q\text{ and }a>0\\p=x+\sqrt{-\frac{q-y}{a}}\text{; }p=x-\sqrt{-\frac{q-y}{a}}\text{, }&y\leq q\text{ and }a<0\\p=x\text{, }&y=q\text{ and }a\neq 0\\p\in \mathrm{R}\text{, }&y=q\text{ and }a=0\end{matrix}\right.$$