<p><a href="http://i.imgur">http://i.imgur</a>. com/3gNtRS4.jpg</p>
<p>Remove the space between . and com to see the image</p>
<p>The answer is E, and the explanation says to complete the square and somehow manipulate this into standard ellipse equation form, but I do not understand how they did it.</p>
<p>Rearrange to:
(3x² - 6x) + (2y² + 4y) -1 = 0</p>
<p>Now to complete the squares. First, remove the coefficient of the ² term from within the brackets:</p>
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<p>Now comes the tricky part. Look at (x² - 2x). Let’s ASSUME it’s of the form a² - 2ab. What we want is (a - b)².</p>
<p>How do we turn (a² - 2ab) into that? First, divide 2ab by 2a. Now it’s (a² - b). Now just move the ² from a to outside the bracket. Now it’s (a - b)². Tada!</p>
<p>Let’s try it with (x² - 2x). Divide the second term by 2a, in this case 2x:
(x² - 1)
Now move the ² outside the bracket:
(x - 1)². That’s the eventual square you want to end up with. But you aren’t there yet.</p>
<p>Go back to the a, b form.
(a - b)² = a² - 2ab + b².
But originally you only had a² - 2ab. So by doing this change, you’ve added b², which you now need to remove from the equation.
So if you turn (x² - 2x) into (x - 1)², you need to REMOVE b², which in this case is (-1)² which is 1.
So all in all, x² - 2x = (x - 1)² - 1</p>
<p>Now let’s try it with (y² + 4y). In this case, a = y. First, divide the second term by 2a, which in this case is 2y:</p>
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<p>So now your original equation looks like this:
3[(x - 1)² -1] + 2[(y + 2)² - 4] - 1 = 0
3(x - 1)² - 3 + 2(y + 2)² - 8 -1 = 0
3(x - 1)² + 2(y + 2)² = 12</p>
<p>Now to turn it into the standard form of an ellipse, divide both sides by 12:
[(x - 1)²]/4 + [(y + 2)²]/6 = 1</p>
<p>Now that it’s in the standard form, it’s easy. The bigger denominator is 6, so y is the major axis. Length of the major axis is twice the square root of 6. (E).</p>
<p>NOTE: The perfect squaring process looks tough at first but it’s really simple:
-Take the coef of the ² term out.
-Divide the other term by 2x.
-Move ² outside the bracket.
-Subtract the square of the second term in the bracket.</p>
<p>Just be careful with + and - signs.</p>