# How do I convert the equation -9x^2 +4y^2 +72x-16y =164 to standard form?

Sep 25, 2014

We need to complete the square for the $x \mathmr{and} {x}^{2}$ terms and then for the $y \mathmr{and} {y}^{2}$ terms.

First, reorder the terms

$- 9 {x}^{2} + 72 x + 4 {y}^{2} - 16 y = 164$

Factor out a $- 9$ from the first 2 terms.
*Remember to change the signs.

Factor out a $4$ from the last 2 terms.

$- 9 \left({x}^{2} - 8 x\right) + 4 \left({y}^{2} - 4 y\right) = 164$

Work with the $x$ term.

${\left(- \frac{8}{2}\right)}^{2} = {\left(- 4\right)}^{2} = 16$

Remember that we factor out $- 9$ so, we have to add $- 9 \cdot 4 = - 36$ to the right side of the equation.

Work with the $y$ term.

${\left(- \frac{4}{2}\right)}^{2} = {\left(- 2\right)}^{2} = 4$

Remember that we factor out $4$ so, we have to add $4 \cdot 4 = 16$ to the right side of the equation.

$- 9 \left({x}^{2} - 8 x + 16\right) + 4 \left({y}^{2} - 4 y + 4\right) = 164 - 144 + 16$

Factor:

${x}^{2} - 8 x + 16 \implies {\left(x - 4\right)}^{2} \implies$ Perfect square trinomial

${y}^{2} - 4 y + 4 \implies {\left(y - 2\right)}^{2} \implies$ Perfect square trinomial

$- 9 {\left(x - 4\right)}^{2} + 4 {\left(y - 2\right)}^{2} = 164 - 144 + 16$

$- 9 {\left(x - 4\right)}^{2} + 4 {\left(y - 2\right)}^{2} = 36$

$\frac{- 9 {\left(x - 4\right)}^{2}}{36} + \frac{4 {\left(y - 2\right)}^{2}}{36} = \frac{36}{36}$

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