Solve (x-p)^2/q+y^2/4=216

$$\frac { ( x - p ) ^ { 2 } } { q } + \frac { y ^ { 2 } } { 4 } = 216$$

Solve for q (complex solution)

$$\left\{\begin{matrix}q=-\frac{4\left(x-p\right)^{2}}{y^{2}-864}\text{, }&x\neq p\text{ and }y\neq -12\sqrt{6}\text{ and }y\neq 12\sqrt{6}\\q\neq 0\text{, }&\left(y=-12\sqrt{6}\text{ or }y=12\sqrt{6}\right)\text{ and }x=p\end{matrix}\right.$$

Steps for Solving Linear Equation

Variable q cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 4q, the least common multiple of q,4.

$$4\left(x-p\right)^{2}+qy^{2}=864q$$

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

$$4\left(x^{2}-2xp+p^{2}\right)+qy^{2}=864q$$

Use the distributive property to multiply 4 by x^{2}-2xp+p^{2}.

$$4x^{2}-8xp+4p^{2}+qy^{2}=864q$$

Subtract 864q from both sides.

$$4x^{2}-8xp+4p^{2}+qy^{2}-864q=0$$

Subtract 4x^{2} from both sides. Anything subtracted from zero gives its negation.

$$-8xp+4p^{2}+qy^{2}-864q=-4x^{2}$$

Add 8xp to both sides.

$$4p^{2}+qy^{2}-864q=-4x^{2}+8xp$$

Subtract 4p^{2} from both sides.

$$qy^{2}-864q=-4x^{2}+8xp-4p^{2}$$

Combine all terms containing q.

$$\left(y^{2}-864\right)q=-4x^{2}+8xp-4p^{2}$$

The equation is in standard form.

$$\left(y^{2}-864\right)q=-4x^{2}+8px-4p^{2}$$

Divide both sides by y^{2}-864.

$$\frac{\left(y^{2}-864\right)q}{y^{2}-864}=-\frac{4\left(x-p\right)^{2}}{y^{2}-864}$$

Dividing by y^{2}-864 undoes the multiplication by y^{2}-864.

$$q=-\frac{4\left(x-p\right)^{2}}{y^{2}-864}$$

Variable q cannot be equal to 0.

$$q=-\frac{4\left(x-p\right)^{2}}{y^{2}-864}\text{, }q\neq 0$$

Solve for q

$$\left\{\begin{matrix}q=-\frac{4\left(x-p\right)^{2}}{y^{2}-864}\text{, }&x\neq p\text{ and }|y|\neq 12\sqrt{6}\\q\neq 0\text{, }&x=p\text{ and }|y|=12\sqrt{6}\end{matrix}\right.$$

Steps for Solving Linear Equation

Variable q cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 4q, the least common multiple of q,4.

$$4\left(x-p\right)^{2}+qy^{2}=864q$$

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

$$4\left(x^{2}-2xp+p^{2}\right)+qy^{2}=864q$$

Use the distributive property to multiply 4 by x^{2}-2xp+p^{2}.

$$4x^{2}-8xp+4p^{2}+qy^{2}=864q$$

Subtract 864q from both sides.

$$4x^{2}-8xp+4p^{2}+qy^{2}-864q=0$$

Subtract 4x^{2} from both sides. Anything subtracted from zero gives its negation.

$$-8xp+4p^{2}+qy^{2}-864q=-4x^{2}$$

Add 8xp to both sides.

$$4p^{2}+qy^{2}-864q=-4x^{2}+8xp$$

Subtract 4p^{2} from both sides.

$$qy^{2}-864q=-4x^{2}+8xp-4p^{2}$$

Combine all terms containing q.

$$\left(y^{2}-864\right)q=-4x^{2}+8xp-4p^{2}$$

The equation is in standard form.

$$\left(y^{2}-864\right)q=-4x^{2}+8px-4p^{2}$$

Divide both sides by -864+y^{2}.

$$\frac{\left(y^{2}-864\right)q}{y^{2}-864}=-\frac{4\left(x-p\right)^{2}}{y^{2}-864}$$

Dividing by -864+y^{2} undoes the multiplication by -864+y^{2}.

$$q=-\frac{4\left(x-p\right)^{2}}{y^{2}-864}$$

Variable q cannot be equal to 0.

$$q=-\frac{4\left(x-p\right)^{2}}{y^{2}-864}\text{, }q\neq 0$$

Solve for p (complex solution)

$$p=-\frac{i\sqrt{q}\sqrt{y^{2}-864}}{2}+x$$

$$p=\frac{i\sqrt{q}\sqrt{y^{2}-864}}{2}+x\text{, }q\neq 0$$

Solve for p

$$\left\{\begin{matrix}p=\frac{\sqrt{864q-qy^{2}}}{2}+x\text{; }p=-\frac{\sqrt{864q-qy^{2}}}{2}+x\text{, }&|y|\geq 12\sqrt{6}\text{ and }q<0\\p=\frac{\sqrt{864q-qy^{2}}}{2}+x\text{; }p=-\frac{\sqrt{864q-qy^{2}}}{2}+x\text{, }&|y|=12\sqrt{6}\text{ and }q\neq 0\\p=\frac{\sqrt{864q-qy^{2}}}{2}+x\text{; }p=-\frac{\sqrt{864q-qy^{2}}}{2}+x\text{, }&q>0\text{ and }|y|\leq 12\sqrt{6}\end{matrix}\right.$$