Can you explain to me what conic section is represented by the equation 3x^2+2y^2-24x+30=0?

The conic section represented by the equation 3x^2 + 2y^2 - 24x + 30 = 0 is an ellipse.

Given, 3x^2 + 2y^2 - 24x + 30 = 0

Complete the square of the equation

3x^2 + 2y^2 - 24x + 30 = 0

3x^2 + 2y^2 - 24x = -30

Group like terms and factorize where possible

3x^2 - 24x + 2y^2 = -30

3(x^2 – 8x) +2y^2 = -30

In order to factor x^2 – 8x as a complete square trinomial, add and subtract 16

3(x^2 – 8x + 16 – 16) + 2y^2 = -30

3((x – 4)^2 – 16) + 2y^2 = -30                                      … Since x^2 – 8x + 16 = (x – 4)^2

3(x – 4)^2 – 3(16) + 2y^2 = -30

3(x – 4)^2 – 48 + 2y^2 = -30

Add 48 to both sides of the equation

3(x – 4)^2 – 48 + 48 + 2y^2 = -30 + 48

3(x – 4)^2 + 2y^2 = 18

Divide both sides by 18

[3(x – 4)^2 + 2y^2]/18 = 18/18

[3(x – 4)^2 + 2y^2]/18 = 1

[3(x – 4)^2]/18 + (2y^2)/18 = 1

[(x – 4)^2]/6 + (y^2)/9 = 1

The general form for the standard equation for an eclipse is [(x^2)/(a^2)] + [(y^2)/(b^2)] = 1

Since the equation, [(x – 4)^2]/6 + (y^2)/9 = 1 can be fitted with the standard form of an ellipse, it is an ellipse.

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