Question

rewrite the equation of the circle x² + y²+2x + 14y+20 = 0 into standard form. enter your answers in the boxes to complete the equation. be sure to include an addition or subtraction symbol where needed. (x )²+(y )²=

Explanation:

Step1: Complete the square for the x - terms

For the $x$ terms in the equation $x^{2}+2x + y^{2}+14y+20 = 0$. In the expression $x^{2}+2x$, we use the formula $(a + b)^2=a^{2}+2ab + b^{2}$. Here $a = x$ and $2ab=2x$, so $b = 1$. Then $x^{2}+2x=(x + 1)^{2}-1$.

Step2: Complete the square for the y - terms

For the $y$ terms $y^{2}+14y$, using the formula $(a + b)^2=a^{2}+2ab + b^{2}$ with $a = y$ and $2ab = 14y$, so $b=7$. Then $y^{2}+14y=(y + 7)^{2}-49$.

Step3: Rewrite the original equation

Substitute the completed - square expressions into the original equation:
\[

$$\begin{align*} (x + 1)^{2}-1+(y + 7)^{2}-49+20&=0\\ (x + 1)^{2}+(y + 7)^{2}-30&=0\\ (x + 1)^{2}+(y + 7)^{2}&=30 \end{align*}$$

\]

Answer:

$(x + 1)^{2}+(y + 7)^{2}=30$