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With the squared terms having opposite signs, this appears to be a hyperbola. So we will attempt to transform the equation into the form:
We start by completing the squares. Most people start this process by moving the constant term to the other side. Adding 164 to each side we get:
Next we gather the x terms and y terms together:
Then we factor out the coefficient of the squared term from each pair of terms:
NOTE: Since we factored out -9 from the x terms, the sign changed in front of the 4x!
Next we need to figure out what number needs to be added to each expression in the parentheses. Take half the coefficient of the first power term and square it. For -4y, half would be -2 and -2 squared would be 4. So we want a 4 to appear in that parentheses. For 4x, half would be 2 and 2 squared is 4. So we want a 4 to appear in the parentheses for the x terms.
The next part is the tricky part. We know that we want 4's to appear in both sets of parentheses. But how do we do this properly? We have to add the same number to both sides. But what number? Let's look at an unfunushed version of what we are trying to do: + ? + ?
What we have to understand is that we are not just adding 4's to the left side. Those 4's are in parentheses and there is a number outside each set of parentheses. When we the first 4, we are really adding 4*4 or 16 because of the 4 in front. And when we add the second 4, we are really adding -9*4 or -36 because of the -9 in front. So to add the 4's properly we need to add 16 and -36 to the right side to balance out what we added to the left side:
which simplifies to:
Now that the expressions in the parentheses are perfect squares we can rewrite them as perfect squares:
Note: If you're not sure where the -2 and +2 came from, they came from the half's we calculated earlier when we were figuring out what we needed to add.
The hardest part is done. Next we get the 1 we want on the right side by dividing both sides by 144:
Fortunately the 4 and the 9 will cancel. (If they didn't we'd still need to make them "disappear" from the numerators.)
A few last details: We want x-h so we rewrite x+2 as an equivalent subtraction. And we want perfect squares in the denominators.
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