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Write the equation in standard form for the ellipse $x^2 + 2y^2 - 6x - 23 = 0$ .

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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 + 2y^2 - 6x - 23 = 0

.

Complete the Square for $x$ : Now, we complete the square for the $x$ terms. To do this, we take the coefficient of the $x$ term, divide it by $2$ , and square it. That's $(-6/2)^2 = 9$ . Add $9$ to both sides. $\newline$ $x^2 - 6x + 9 + 2y^2 = 23 + 9$
Factor out $2$ for $y$ : We do the same for the $y$ terms, but since there's a coefficient of $2$ in front of $y^2$ , we need to factor that out first. $\newline$ $2(y^2) = 2(y^2 + 0y + 0)$ $\newline$ We don't need to complete the square for $y$ because there's no $y$ term to complete the square with. $\newline$ So, we just have $2(y^2)$ .
Rewrite with Completed Square: Now we rewrite the equation with the completed square for $x$ and the $y$ term. $\newline$ $(x - 3)^2 + 2(y^2) = 32$
Divide by $32$ : Divide everything by $32$ to get the standard form of the ellipse. $(x - 3)^2/32 + 2(y^2)/32 = 1$
Simplify $y$ Term: Simplify the $y$ term by dividing the coefficient $2$ by $32$ . $\left(x - 3\right)^2/32 + \left(y^2\right)/16 = 1$

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