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Write the equation in standard form for the ellipse $x^2 + 9y^2 + 2x - 26 = 0$ .

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Convert equations of ellipses from general to standard form

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Q. Write the equation in standard form for the ellipse

x^2 + 9y^2 + 2x - 26 = 0

.

Complete the Square for $x$ : Now, we complete the square for the $x$ terms. To do this, we take half of the coefficient of $x$ , square it, and add it to both sides. $\newline$ Half of $2$ is $1$ , and $1$ squared is $1$ , so we add $1$ to both sides. $\newline$ $x^2 + 2x + 1 + 9y^2 = 26 + 1$
Simplify After Adding $1$ : Simplify the equation after adding $1$ to both sides. $\newline$ $(x + 1)^2 + 9y^2 = 27$
Complete the Square for $y$ : Now, we need to complete the square for the $y$ terms, but since there's no linear $y$ term, we don't need to add anything for $y$ . $\newline$ So, we just rewrite the $y$ term as it is. $\newline$ $(x + 1)^2 + 9(y)^2 = 27$
Divide by $27$ : Finally, we divide the entire equation by $27$ to get the standard form of the ellipse. $(x + 1)^2/27 + 9(y)^2/27 = 27/27$
Simplify by Dividing: Simplify the equation by dividing each term by $27$ . $\frac{(x + 1)^2}{27} + \frac{(y)^2}{3} = 1$

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