Solve y=a(x-4)^2-3

$$y = a ( x - 4 ) ^ { 2 } - 3$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=\frac{y+3}{\left(x-4\right)^{2}}\text{, }&x\neq 4\\a\in \mathrm{C}\text{, }&y=-3\text{ and }x=4\end{matrix}\right.$$

Steps for Solving Linear Equation

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-4\right)^{2}.

$$y=a\left(x^{2}-8x+16\right)-3$$

Use the distributive property to multiply a by x^{2}-8x+16.

$$y=ax^{2}-8ax+16a-3$$

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

$$ax^{2}-8ax+16a-3=y$$

Add 3 to both sides.

$$ax^{2}-8ax+16a=y+3$$

Combine all terms containing a.

$$\left(x^{2}-8x+16\right)a=y+3$$

Divide both sides by x^{2}-8x+16.

$$\frac{\left(x^{2}-8x+16\right)a}{x^{2}-8x+16}=\frac{y+3}{x^{2}-8x+16}$$

Dividing by x^{2}-8x+16 undoes the multiplication by x^{2}-8x+16.

$$a=\frac{y+3}{x^{2}-8x+16}$$

Divide y+3 by x^{2}-8x+16.

$$a=\frac{y+3}{\left(x-4\right)^{2}}$$

Solve for a

$$\left\{\begin{matrix}a=\frac{y+3}{\left(x-4\right)^{2}}\text{, }&x\neq 4\\a\in \mathrm{R}\text{, }&y=-3\text{ and }x=4\end{matrix}\right.$$

Steps for Solving Linear Equation

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-4\right)^{2}.

$$y=a\left(x^{2}-8x+16\right)-3$$

Use the distributive property to multiply a by x^{2}-8x+16.

$$y=ax^{2}-8ax+16a-3$$

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

$$ax^{2}-8ax+16a-3=y$$

Add 3 to both sides.

$$ax^{2}-8ax+16a=y+3$$

Combine all terms containing a.

$$\left(x^{2}-8x+16\right)a=y+3$$

Divide both sides by x^{2}-8x+16.

$$\frac{\left(x^{2}-8x+16\right)a}{x^{2}-8x+16}=\frac{y+3}{x^{2}-8x+16}$$

Dividing by x^{2}-8x+16 undoes the multiplication by x^{2}-8x+16.

$$a=\frac{y+3}{x^{2}-8x+16}$$

Divide y+3 by x^{2}-8x+16.

$$a=\frac{y+3}{\left(x-4\right)^{2}}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=4+a^{-\frac{1}{2}}\sqrt{y+3}\text{; }x=4-a^{-\frac{1}{2}}\sqrt{y+3}\text{, }&a\neq 0\\x\in \mathrm{C}\text{, }&y=-3\text{ and }a=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=-\sqrt{\frac{y+3}{a}}+4\text{; }x=\sqrt{\frac{y+3}{a}}+4\text{, }&y\leq -3\text{ and }a<0\\x=\frac{-\sqrt{y+3}+4\sqrt{a}}{\sqrt{a}}\text{; }x=\frac{\sqrt{y+3}+4\sqrt{a}}{\sqrt{a}}\text{, }&y\geq -3\text{ and }a>0\\x\in \mathrm{R}\text{, }&y=-3\text{ and }a=0\end{matrix}\right.$$