Complete the square in 25x^2-4y^2+100x+24y-36=0
- Thread starter CaliMan
- Start date
Re: Please help Complete the square in a hyperbola
Hello, CaliMan!
Factor: \(\displaystyle \:25(x^2\,+\,4x\;\;\
\,-\,4(y^2\,-\,6y\;\;\\:=\:36\)
Complete the square:
. . . . . \(\displaystyle 25(x^2\,+\,4x\,\)+ 4\(\displaystyle )\,-\,4(y^2\,-\,6y\,\)+ 9\(\displaystyle ) \;=\;36\,\)+ 100\(\displaystyle \,\) - 36 . **
Factor: \(\displaystyle \:25(x\,+\,2)^2\,-\,4(y\,-\,3)^2\:=\:100\)
Divide by 100: \(\displaystyle \L\:\fbox{\frac{(x\,+\,2)^2}{4}\,-\,\frac{(y\,-\,3)^2}{25}\;=\;1}\)
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** .Important
We wrote + 4 on the left side.
. . But it is multiplied by the "25" out front.
So we actually added 100 to the left side.
. . Hence, we add 100 to the right side.
We wrote + 9 on the left side.
. . But it is multiplied by the "-4" out front.
So we actually subtracted 36 from the left side.
. . Hence, we subtract 36 from the right side.
Hello, CaliMan!
\(\displaystyle 25x^2\,-\,4y^2\,+\,100x\,+\,24y\,-\,36\:=\:0\)
I've gotten to this point: \(\displaystyle \
Factor: \(\displaystyle \:25(x^2\,+\,4x\;\;\
\,-\,4(y^2\,-\,6y\;\;\\:=\:36\)Complete the square:
. . . . . \(\displaystyle 25(x^2\,+\,4x\,\)+ 4\(\displaystyle )\,-\,4(y^2\,-\,6y\,\)+ 9\(\displaystyle ) \;=\;36\,\)+ 100\(\displaystyle \,\) - 36 . **
Factor: \(\displaystyle \:25(x\,+\,2)^2\,-\,4(y\,-\,3)^2\:=\:100\)
Divide by 100: \(\displaystyle \L\:\fbox{\frac{(x\,+\,2)^2}{4}\,-\,\frac{(y\,-\,3)^2}{25}\;=\;1}\)
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
** .Important
We wrote + 4 on the left side.
. . But it is multiplied by the "25" out front.
So we actually added 100 to the left side.
. . Hence, we add 100 to the right side.
We wrote + 9 on the left side.
. . But it is multiplied by the "-4" out front.
So we actually subtracted 36 from the left side.
. . Hence, we subtract 36 from the right side.