Question

The equation of a circle in general form is x squared plus y squared plus 6 x minus 4 y plus 4 equals 0.

Write the equation in standard form.

(1 point)
Responses

left-parenthesis x plus 3 right-parenthesis squared plus left-parenthesis y minus 2 right-parenthesis squared equals 4
Image with alt text: left-parenthesis x plus 3 right-parenthesis squared plus left-parenthesis y minus 2 right-parenthesis squared equals 4

left-parenthesis x plus 3 right-parenthesis squared plus left-parenthesis y minus 2 right-parenthesis squared equals 9
Image with alt text: left-parenthesis x plus 3 right-parenthesis squared plus left-parenthesis y minus 2 right-parenthesis squared equals 9

left-parenthesis x plus 3 right-parenthesis squared plus left-parenthesis y plus 2 right-parenthesis squared equals 4
Image with alt text: left-parenthesis x plus 3 right-parenthesis squared plus left-parenthesis y plus 2 right-parenthesis squared equals 4

left-parenthesis x plus 3 right-parenthesis squared plus left-parenthesis y plus 2 right-parenthesis squared equals 9
Image with alt text: left-parenthesis x plus 3 right-parenthesis squared plus left-parenthesis y plus 2 right-parenthesis squared equals 9

Answer 1

To convert the given equation of the circle from general form to standard form, we start with the equation:

\[ x^2 + y^2 + 6x - 4y + 4 = 0. \]

First, we reorganize it:

\[ x^2 + 6x + y^2 - 4y + 4 = 0. \]

Next, we move the constant term to the other side:

\[ x^2 + 6x + y^2 - 4y = -4. \]

Now we will complete the square for both \(x\) and \(y\).

**For \(x^2 + 6x\):**
1. Take the coefficient of \(x\), which is 6, divide by 2 to get 3, and square it to get 9.
2. Thus, we can write:
\[ x^2 + 6x = (x + 3)^2 - 9. \]

**For \(y^2 - 4y\):**
1. Take the coefficient of \(y\), which is -4, divide by 2 to get -2, and square it to get 4.
2. Thus, we can write:
\[ y^2 - 4y = (y - 2)^2 - 4. \]

Substituting these back into the equation:

\[(x + 3)^2 - 9 + (y - 2)^2 - 4 = -4.\]

Simplifying this gives:

\[(x + 3)^2 + (y - 2)^2 - 13 = -4,\]

which simplifies to:

\[(x + 3)^2 + (y - 2)^2 = 9.\]

Now the equation is in standard form:

\[(x + 3)^2 + (y - 2)^2 = 9.\]

Thus, the correct response is:

**left-parenthesis x plus 3 right-parenthesis squared plus left-parenthesis y minus 2 right-parenthesis squared equals 9.**