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Write the equation in standard form for the ellipse $2x^2 + y^2 + 8x + 4 = 0$ .

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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

2x^2 + y^2 + 8x + 4 = 0

.

Complete the Square for $x$ Terms: Now, we complete the square for the $x$ terms. To do this, we take the coefficient of the $x$ term, which is $8$ , divide it by $2$ , and square it. That's $(8/2)^2 = 16$ . We add and subtract this inside the equation to maintain equality. $\newline$ $2(x^2 + 4x + 16) - 32 + y^2 = -4$
Complete the Square for $y$ Terms: Next, we complete the square for the $y$ terms. Since there's no $y$ term to complete the square with, we just leave the $y^2$ as it is. $\newline$ $2(x^2 + 4x + 16) - 32 + y^2 = -4$
Factor Completed Square for x Terms: Now, we factor the completed square for the x terms. $2(x + 2)^2 - 32 + y^2 = -4$
Isolate $y^2$ : We then add $32$ to both sides to isolate the $y^2$ on one side. $\newline$ $2(x + 2)^2 + y^2 = 28$
Divide by $28$ : Finally, we divide the entire equation by $28$ to get the standard form of the ellipse. $(x + 2)^2/14 + y^2/28 = 1$

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