Final answer:
To complete the equation of the ellipse x²+8x+3y²-6y+7=0, complete the square for the x and y terms and rewrite the equation in standard form, resulting in (x+4)²/9 + (y-1)²/3 = 1.
Step-by-step explanation:
The student has asked to complete the equation of an ellipse. The given equation is x²+8x+3y²-6y+7=0.
To complete the equation, we first need to group the x and y terms and then complete the square for each group:
Group x terms: x²+8x, and add (8/2)² = 16 to complete the square.
Group y terms: 3y²-6y, and add (6/2∙3)² = 1 to complete the square.
After completing the square, subtract the added values on the right side of the equation to maintain equality.
Here's how it looks when we complete the square:
For x terms: x² + 8x + 16 - 16
For y terms: 3(y² - 2y + 1) - 3
Combine the completed squares with the constant term: (x+4)² + 3(y-1)² - 16 + 7 = 0
Rewrite the equation: (x+4)² + 3(y-1)² = 9
Divide by 9 to normalize the right side to 1: (x+4)²/9 + (y-1)²/3 = 1
The completed equation of the ellipse in standard form is (x+4)²/9 + (y-1)²/3 = 1. This is the canonical form of the equation for an ellipse, which enables us to identify its key properties, such as center, axes lengths, and foci.