Solve y=-aleft(x-hright)^2+k

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

Solve for a (complex solution)

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

Steps for Solving Linear Equation

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

$$y=\left(-a\right)\left(x^{2}-2xh+h^{2}\right)+k$$

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

$$y=\left(-a\right)x^{2}-2\left(-a\right)xh+\left(-a\right)h^{2}+k$$

Multiply -2 and -1 to get 2.

$$y=\left(-a\right)x^{2}+2axh+\left(-a\right)h^{2}+k$$

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

$$\left(-a\right)x^{2}+2axh+\left(-a\right)h^{2}+k=y$$

Subtract k from both sides.

$$\left(-a\right)x^{2}+2axh+\left(-a\right)h^{2}=y-k$$

Reorder the terms.

$$-ax^{2}+2ahx-ah^{2}=y-k$$

Combine all terms containing a.

$$\left(-x^{2}+2hx-h^{2}\right)a=y-k$$

Divide both sides by -x^{2}+2hx-h^{2}.

$$\frac{\left(-x^{2}+2hx-h^{2}\right)a}{-x^{2}+2hx-h^{2}}=\frac{y-k}{-x^{2}+2hx-h^{2}}$$

Dividing by -x^{2}+2hx-h^{2} undoes the multiplication by -x^{2}+2hx-h^{2}.

$$a=\frac{y-k}{-x^{2}+2hx-h^{2}}$$

Divide y-k by -x^{2}+2hx-h^{2}.

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

Solve for a

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

Steps for Solving Linear Equation

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

$$y=\left(-a\right)\left(x^{2}-2xh+h^{2}\right)+k$$

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

$$y=\left(-a\right)x^{2}-2\left(-a\right)xh+\left(-a\right)h^{2}+k$$

Multiply -2 and -1 to get 2.

$$y=\left(-a\right)x^{2}+2axh+\left(-a\right)h^{2}+k$$

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

$$\left(-a\right)x^{2}+2axh+\left(-a\right)h^{2}+k=y$$

Subtract k from both sides.

$$\left(-a\right)x^{2}+2axh+\left(-a\right)h^{2}=y-k$$

Reorder the terms.

$$-ax^{2}+2ahx-ah^{2}=y-k$$

Combine all terms containing a.

$$\left(-x^{2}+2hx-h^{2}\right)a=y-k$$

Divide both sides by -x^{2}+2hx-h^{2}.

$$\frac{\left(-x^{2}+2hx-h^{2}\right)a}{-x^{2}+2hx-h^{2}}=\frac{y-k}{-x^{2}+2hx-h^{2}}$$

Dividing by -x^{2}+2hx-h^{2} undoes the multiplication by -x^{2}+2hx-h^{2}.

$$a=\frac{y-k}{-x^{2}+2hx-h^{2}}$$

Divide y-k by -x^{2}+2hx-h^{2}.

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

Solve for h (complex solution)

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

Solve for h

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