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Write the equation in standard form for the ellipse $x^2 + 7y^2 + 14y - 7 = 0$ .

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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 + 7y^2 + 14y - 7 = 0

.

Complete the Square for $y$ : Now, we need to complete the square for the $y$ terms. To do this, we add and subtract $\left(\frac{b}{2}\right)^2$ where $b$ is the coefficient of $y$ . In this case, $b$ is $2$ , so we add and subtract $\left(\frac{2}{2}\right)^2 = 1$ inside the parentheses. $\newline$ $x^2 + 7(y^2 + 2y + 1 - 1) = 7$
Combine Like Terms: Simplify the equation by combining like terms inside the parentheses. $x^2 + 7((y + 1)^2 - 1) = 7$
Distribute $7$ : Distribute the $7$ to both terms inside the parentheses. $x^2 + 7(y + 1)^2 - 7 = 7$
Isolate Terms with Variables: Add $7$ to both sides to isolate the terms with variables on one side. $\newline$ $x^2 + 7(y + 1)^2 = 14$
Divide by $14$ : Divide the entire equation by $14$ to get the standard form of the ellipse. $\frac{x^2}{14} + \frac{7(y + 1)^2}{14} = 1$
Simplify Fractions: Simplify the fractions. $\frac{x^2}{14} + \frac{(y + 1)^2}{2} = 1$

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