Write the equation in standard form for the circle $x^2+y^2+6x-1= 0$ .

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Algebra 2

Convert equations of circles from general to standard form

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Q. Write the equation in standard form for the circle

x^2+y^2+6x-1= 0

Identify equation and need: Identify the given equation and the need to complete the square for the $x$ -terms. $\newline$ The given equation is $x^2 + y^2 + 6x - 1 = 0$ . To write the equation in standard form, we need to complete the square for the $x$ -terms.
Group x-terms and leave space: Group the x-terms together and leave a space to complete the square. $\newline$ $x^2 + 6x + (\ ) + y^2 = 1$ $\newline$ We have moved the constant term to the other side by adding $1$ to both sides.
Find number to complete square: Find the number to complete the square for the $x$ -terms. $\newline$ To complete the square, we take half of the coefficient of $x$ , which is $\frac{6}{2} = 3$ , and square it, giving us $3^2 = 9$ .
Add and subtract inside parentheses: Add and subtract the number found in Step $3$ inside the parentheses. $\newline$ $x^2 + 6x + 9 - 9 + y^2 = 1$ $\newline$ We added $9$ to complete the square and then subtracted $9$ to keep the equation balanced.
Rewrite equation with completed square: Rewrite the equation with the completed square for the x-terms. $\newline$ $(x + 3)^2 - 9 + y^2 = 1$ $\newline$ We have now completed the square for the x-terms.
Move constant term from completed square: Move the constant term from the completed square to the other side. $\newline$ $(x + 3)^2 + y^2 = 1 + 9$ $\newline$ We added $9$ to both sides to isolate the completed square and $y^2$ on one side.
Simplify right side of equation: Simplify the right side of the equation. $\newline$ $(x + 3)^2 + y^2 = 10$ $\newline$ We have combined the constants on the right side.
Write final equation in standard form: Write the final equation in standard form. $\newline$ The standard form for the equation of a circle is $(x - h)^2 + (y - k)^2 = r^2$ , where $(h, k)$ is the center and $r$ is the radius. Our equation is now in this form with $h = -3$ , $k = 0$ , and $r^2 = 10$ .