drag the signs and values to the correct locations on the image. each sign and value can be used more than once, but not all signs and values will be used.
complete the standard form of the equation of the ellipse represented by the equation 9x2 + 4y2 − 36x + 8y + 4 = 0.

(x - 2)²/2² + (y + 1)²/3² = 1 ⇒ The bold values and signs are the answers

Step-by-step explanation:

* Lets revise the equation of the ellipse  

- The standard form of the equation of an ellipse with center (h , k)  

 and major axis parallel to x-axis is (x - h)²/a² + (y - k)²/b² = 1  

- The coordinates of the vertices are (h ± a , k)

- To change the form of the equation of the ellipse to standard form we

 will using the completing square

∵ The equation of the ellipse is 9x² + 4y² - 36x + 8y + 4 = 0

- Lets collect x in bracket and y in bracket

∴ (9x² - 36x) + (4y² + 8y) + 4 = 0

- We will take a common factor 9 from the bracket of x and 4 from the

 bracket of y

∴ 9(x² - 4x) + 4(y² + 2y) + 4 = 0

- Lets make 9(x² - 4x) a completing square

∵ √x² = x ⇒ the 1st term in the bracket

∵ 4x ÷ 2 = 2x ⇒ the product of the 1st and 2nd terms

∵ 2x ÷ x = 2 ⇒ the 2nd term in the bracket

∴ The bracket is (x - 2)²

∵ (x - 2)² = x² - 4x + 4 ⇒ we will add 4 in the bracket and subtract 4

  out the bracket

∴ 9[(x² - 4x + 4) - 4] = 9[(x - 2)² - 4]

- Lets make 4(y² + 2y) a completing square

∵ √y² = y ⇒ the 1st term in the bracket

∵ 2y ÷ 2 = y ⇒ the product of the 1st and 2nd terms

∵ y ÷ y = 1 ⇒ the 2nd term in the bracket

∴ The bracket is (y + 1)²

∵ (y + 1)² = y² - 2y + 1 ⇒ we will add 1 in the bracket and subtract 1

  out the bracket

∴ 4[(y² + 2y + 1) - 1] = 4[(y + 1)² - 1]

- Lets write the equation with the completing square

∴ 9[(x - 2)² - 4] + 4[(y + 1)² - 1] + 4 = 0 ⇒ simplify

∴ 9(x -2)² - 36 + 4(y + 1)² - 4 + 4 = 0 ⇒ add the numerical terms

∴ 9(x - 2)² + 4(y + 1)² - 36 = 0 ⇒ add 36 to both sides

∴ 9(x - 2)² + 4(y + 1)² = 36 ⇒ divide both sides by 36

∴ (x - 2)²/4 + (y + 1)²/9 = 1

∵ 4 = 2² and 9 = 3²

∴ (x - 2)²/2² + (y + 1)²/3² = 1

* The standard form of the equation of the ellipse is

  (x - 2)²/2² + (y + 1)²/3² = 1