Solve (x)=a(x-h)^2+k

$$( x ) = a ( x - h ) ^ { 2 } + k$$

Solve for a (complex solution)

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

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}.

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

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

$$x=ax^{2}-2axh+ah^{2}+k$$

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

$$ax^{2}-2axh+ah^{2}+k=x$$

Subtract k from both sides.

$$ax^{2}-2axh+ah^{2}=x-k$$

Combine all terms containing a.

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

The equation is in standard form.

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

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

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

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

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

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

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

Solve for a

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

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}.

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

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

$$x=ax^{2}-2axh+ah^{2}+k$$

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

$$ax^{2}-2axh+ah^{2}+k=x$$

Subtract k from both sides.

$$ax^{2}-2axh+ah^{2}=x-k$$

Combine all terms containing a.

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

The equation is in standard form.

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

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

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

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

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

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

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

Solve for h (complex solution)

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

Solve for h

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