Solve 6n9x^2+25y^2-54x+100y-44=0

$$6 n 9 x ^ { 2 } + 25 y ^ { 2 } - 54 x + 100 y - 44 = 0$$

Solve for n_9 (complex solution)

$$\left\{\begin{matrix}n_{9}=-\frac{25y^{2}-54x+100y-44}{6x^{2}}\text{, }&x\neq 0\\n_{9}\in \mathrm{C}\text{, }&\left(y=\frac{2}{5}\text{ or }y=-\frac{22}{5}\right)\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract 25y^{2} from both sides. Anything subtracted from zero gives its negation.

$$6n_{9}x^{2}-54x+100y-44=-25y^{2}$$

Add 54x to both sides.

$$6n_{9}x^{2}+100y-44=-25y^{2}+54x$$

Subtract 100y from both sides.

$$6n_{9}x^{2}-44=-25y^{2}+54x-100y$$

Add 44 to both sides.

$$6n_{9}x^{2}=-25y^{2}+54x-100y+44$$

The equation is in standard form.

$$6x^{2}n_{9}=54x-25y^{2}-100y+44$$

Divide both sides by 6x^{2}.

$$\frac{6x^{2}n_{9}}{6x^{2}}=\frac{54x-25y^{2}-100y+44}{6x^{2}}$$

Dividing by 6x^{2} undoes the multiplication by 6x^{2}.

$$n_{9}=\frac{54x-25y^{2}-100y+44}{6x^{2}}$$

Solve for n_9

$$\left\{\begin{matrix}n_{9}=-\frac{25y^{2}-54x+100y-44}{6x^{2}}\text{, }&x\neq 0\\n_{9}\in \mathrm{R}\text{, }&\left(y=-\frac{22}{5}\text{ or }y=\frac{2}{5}\right)\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract 25y^{2} from both sides. Anything subtracted from zero gives its negation.

$$6n_{9}x^{2}-54x+100y-44=-25y^{2}$$

Add 54x to both sides.

$$6n_{9}x^{2}+100y-44=-25y^{2}+54x$$

Subtract 100y from both sides.

$$6n_{9}x^{2}-44=-25y^{2}+54x-100y$$

Add 44 to both sides.

$$6n_{9}x^{2}=-25y^{2}+54x-100y+44$$

The equation is in standard form.

$$6x^{2}n_{9}=54x-25y^{2}-100y+44$$

Divide both sides by 6x^{2}.

$$\frac{6x^{2}n_{9}}{6x^{2}}=\frac{54x-25y^{2}-100y+44}{6x^{2}}$$

Dividing by 6x^{2} undoes the multiplication by 6x^{2}.

$$n_{9}=\frac{54x-25y^{2}-100y+44}{6x^{2}}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{3\left(243+88n_{9}-200n_{9}y-50n_{9}y^{2}\right)}+27}{6n_{9}}\text{; }x=\frac{-\sqrt{3\left(243+88n_{9}-200n_{9}y-50n_{9}y^{2}\right)}+27}{6n_{9}}\text{, }&n_{9}\neq 0\\x=\frac{25y^{2}+100y-44}{54}\text{, }&n_{9}=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{3\left(243+88n_{9}-200n_{9}y-50n_{9}y^{2}\right)}+27}{6n_{9}}\text{; }x=\frac{-\sqrt{3\left(243+88n_{9}-200n_{9}y-50n_{9}y^{2}\right)}+27}{6n_{9}}\text{, }&\left(n_{9}\neq 0\text{ and }y=-\frac{22}{5}\right)\text{ or }\left(n_{9}\neq 0\text{ and }y\leq -\frac{22}{5}\text{ and }n_{9}\leq -\frac{729}{264-600y-150y^{2}}\right)\text{ or }\left(n_{9}\neq 0\text{ and }y=\frac{2}{5}\right)\text{ or }\left(n_{9}\neq 0\text{ and }y\geq \frac{2}{5}\text{ and }n_{9}\leq -\frac{729}{264-600y-150y^{2}}\right)\text{ or }\left(n_{9}\neq 0\text{ and }n_{9}\geq -\frac{729}{264-600y-150y^{2}}\text{ and }y\geq -\frac{22}{5}\text{ and }y\leq \frac{2}{5}\right)\text{ or }\left(y\neq \frac{2}{5}\text{ and }y\neq -\frac{22}{5}\text{ and }n_{9}=-\frac{729}{264-600y-150y^{2}}\right)\\x=\frac{25y^{2}+100y-44}{54}\text{, }&n_{9}=0\end{matrix}\right.$$