uation of the circle is $x^{2}+18 x+y^{2}-12 y-27=0$ . $\newline$ the standard form equation of a circle. Show your reasoning.

Full solution

Q. uation of the circle is

x^{2}+18 x+y^{2}-12 y-27=0

\newline

the standard form equation of a circle. Show your reasoning.

Group x's and y's: Group x's and y's together and move the constant to the other side. $\newline$ $x^2 + 18x + y^2 - 12y = 27$
Complete $x$ square: Complete the square for the $x$ terms. $\newline$ $x^2 + 18x + \left(\frac{18}{2}\right)^2 - \left(\frac{18}{2}\right)^2$ $\newline$ Add and subtract $\left(\frac{18}{2}\right)^2$ inside the equation.
Complete $y$ square: Complete the square for the $y$ terms. $\newline$ $y^2 - 12y + \left(\frac{12}{2}\right)^2 - \left(\frac{12}{2}\right)^2$ $\newline$ Add and subtract $\left(\frac{12}{2}\right)^2$ inside the equation.
Add squares: Add the squares to both sides to keep the equation balanced. $\newline$ $x^2 + 18x + 81 - 81 + y^2 - 12y + 36 - 36 = 27$
Group trinomials: Group the perfect square trinomials and the constants. $\newline$ $(x^2 + 18x + 81) - 81 + (y^2 - 12y + 36) - 36 = 27$
Factor trinomials: Factor the perfect square trinomials. $\newline$ $(x + 9)^2 - 81 + (y - 6)^2 - 36 = 27$
Move constants: Move the constants to the other side by adding them to $27$ . $\newline$ $(x + 9)^2 + (y - 6)^2 = 27 + 81 + 36$
Calculate sum: Calculate the sum of the constants. $27 + 81 + 36 = 144$