Solve 4EX^2+(y-5)^2=16

$$4 E X ^ { 2 } + ( y - 5 ) ^ { 2 } = 16$$

Solve for E (complex solution)

$$\left\{\begin{matrix}E=-\frac{\left(y-9\right)\left(y-1\right)}{4X^{2}}\text{, }&X\neq 0\\E\in \mathrm{C}\text{, }&\left(y=1\text{ or }y=9\right)\text{ and }X=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(y-5\right)^{2}.

$$4EX^{2}+y^{2}-10y+25=16$$

Subtract y^{2} from both sides.

$$4EX^{2}-10y+25=16-y^{2}$$

Add 10y to both sides.

$$4EX^{2}+25=16-y^{2}+10y$$

Subtract 25 from both sides.

$$4EX^{2}=16-y^{2}+10y-25$$

Subtract 25 from 16 to get -9.

$$4EX^{2}=-9-y^{2}+10y$$

The equation is in standard form.

$$4X^{2}E=-y^{2}+10y-9$$

Divide both sides by 4X^{2}.

$$\frac{4X^{2}E}{4X^{2}}=\frac{\left(1-y\right)\left(y-9\right)}{4X^{2}}$$

Dividing by 4X^{2} undoes the multiplication by 4X^{2}.

$$E=\frac{\left(1-y\right)\left(y-9\right)}{4X^{2}}$$

Solve for E

$$\left\{\begin{matrix}E=-\frac{\left(y-9\right)\left(y-1\right)}{4X^{2}}\text{, }&X\neq 0\\E\in \mathrm{R}\text{, }&\left(y=1\text{ or }y=9\right)\text{ and }X=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(y-5\right)^{2}.

$$4EX^{2}+y^{2}-10y+25=16$$

Subtract y^{2} from both sides.

$$4EX^{2}-10y+25=16-y^{2}$$

Add 10y to both sides.

$$4EX^{2}+25=16-y^{2}+10y$$

Subtract 25 from both sides.

$$4EX^{2}=16-y^{2}+10y-25$$

Subtract 25 from 16 to get -9.

$$4EX^{2}=-9-y^{2}+10y$$

The equation is in standard form.

$$4X^{2}E=-y^{2}+10y-9$$

Divide both sides by 4X^{2}.

$$\frac{4X^{2}E}{4X^{2}}=\frac{\left(1-y\right)\left(y-9\right)}{4X^{2}}$$

Dividing by 4X^{2} undoes the multiplication by 4X^{2}.

$$E=\frac{\left(1-y\right)\left(y-9\right)}{4X^{2}}$$

Solve for X (complex solution)

$$\left\{\begin{matrix}X=-\frac{iE^{-\frac{1}{2}}\sqrt{y-9}\sqrt{y-1}}{2}\text{; }X=\frac{iE^{-\frac{1}{2}}\sqrt{y-9}\sqrt{y-1}}{2}\text{, }&E\neq 0\\X\in \mathrm{C}\text{, }&\left(y=1\text{ or }y=9\right)\text{ and }E=0\end{matrix}\right.$$

Solve for X

$$\left\{\begin{matrix}X=\frac{\sqrt{\frac{\left(1-y\right)\left(y-9\right)}{E}}}{2}\text{; }X=-\frac{\sqrt{\frac{\left(1-y\right)\left(y-9\right)}{E}}}{2}\text{, }&\left(y\leq 1\text{ and }E<0\right)\text{ or }\left(y\geq 9\text{ and }E<0\right)\text{ or }\left(y\leq 9\text{ and }y\geq 1\text{ and }E>0\right)\\X\in \mathrm{R}\text{, }&\left(y=1\text{ or }y=9\right)\text{ and }E=0\end{matrix}\right.$$