# How do I graph 9x^2+16y^2-36x+32y=92 algebraically?

Oct 19, 2014

By grouping $x$'s and $y$'s together,

$R i g h t a r r o w \left(9 {x}^{2} - 36 x\right) + \left(16 {y}^{2} + 32 y\right) = 92$

by factoring out 9 and 16,

$R i g h t a r r o w 9 \left({x}^{2} - 4 x\right) + 16 \left({y}^{2} + 2 y\right) = 92$

by adding and subtracting 4 and 1 in the parentheses,

$R i g h t a r r o w 9 \left({x}^{2} - 4 x + 4 - 4\right) + 16 \left({y}^{2} + 2 y + 1 - 1\right) = 92$

by keeping the first three terms in each group,

$R i g h t a r r o w 9 \left({x}^{2} - 4 x + 4\right) - 36 + 16 \left({y}^{2} + 2 y + 1\right) - 16 = 92$

by completing the squares and adding 36 and 16,

$R i g h t a r r o w 9 {\left(x - 2\right)}^{2} + 16 {\left(y + 1\right)}^{2} = 144$

by dividing by 144,

$R i g h t a r r o w {\left(x - 2\right)}^{2} / \left\{16\right\} + {\left(y + 1\right)}^{2} / \left\{9\right\} = 1$

Hence, we have an equation of an ellipse

${\left(x - 2\right)}^{2} / \left\{{4}^{2}\right\} + {\left(y + 1\right)}^{2} / \left\{{3}^{2}\right\} = 1$,

which is 8 units wide, 6 units tall, and centered at $\left(2 , - 1\right)$. I hope that this was helpful.