Godlovesme
  • Godlovesme
HELPPPPPP ASAP PLEASE!! D: Using the following equation, find the center and radius of the circle. x2 + 2x + y2 + 4y = 20
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
acxbox22
  • acxbox22
http://www.regentsprep.org/regents/math/algtrig/atc1/circlelesson.htm
acxbox22
  • acxbox22
sorry i gtg ^ that links explains what you have to do
Godlovesme
  • Godlovesme
awh man :c

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Godlovesme
  • Godlovesme
its ok i'll try to figure it out thanks tho :)
Godlovesme
  • Godlovesme
@pooja195 help me please :((
asnaseer
  • asnaseer
you need to transform this equation:\[x^2 + 2x + y^2 + 4y = 20\]into the standard form:\[(x-h)^2+(y-k)^2=r^2\]which gives a circle of radius \(r\) and center at \((h,k)\)
asnaseer
  • asnaseer
start by looking at just the terms involving \(x\), i.e.:\[x^2+2x\]can you transform this into a form like \((x+?)^2\)?
Godlovesme
  • Godlovesme
i'll try, sorry im not good at this
acxbox22
  • acxbox22
im back
asnaseer
  • asnaseer
we know that:\[(x+a)^2=x^2+2ax+a^2\]
asnaseer
  • asnaseer
and we want to get:\[x^2+2x\]
acxbox22
  • acxbox22
well like the website says, you need to transform it into the center radius form by completing the square
asnaseer
  • asnaseer
can you see what value of \(a\) we need to pick?
acxbox22
  • acxbox22
nvr mind i will let @asnaseer explain :P
Godlovesme
  • Godlovesme
what do u mean @asnaseer
Godlovesme
  • Godlovesme
thanks though :)) @acxbox22
asnaseer
  • asnaseer
we know that:\[(x+a)^2=x^2+2ax+a^2\]can you see what value of \(a\) we need to pick so that we get a \(2x\) in the expansion on the right-hand-side?
Godlovesme
  • Godlovesme
is it 1?
asnaseer
  • asnaseer
correct! :)
Godlovesme
  • Godlovesme
phew than God i felt so stupid lol
asnaseer
  • asnaseer
so we know that:\[(x+1)^2=x^2+2x+1\]agreed so far?
Godlovesme
  • Godlovesme
yes :)
asnaseer
  • asnaseer
but we need just \(x^2+2x\), so we need to subtract 1 from both sides to get:\[(x+1)^2-1=x^2+2x\]agreed?
asnaseer
  • asnaseer
i.e. we know:\[(x+1)^2=x^2+2x+1\]\[\therefore (x+1)^2-1=x^2+2x+1-1=x^2+2x\]
Godlovesme
  • Godlovesme
ohhh so now we're left with \[x^2+2x\] ?
asnaseer
  • asnaseer
exactly
asnaseer
  • asnaseer
now remember your original equation was:\[x^2 + 2x + y^2 + 4y = 20\]we can now substitute \(x^2+2x\) with the new expression we found to get:\[(x+1)^2-1+y^2+4y=20\]understand?
Godlovesme
  • Godlovesme
yes :)
asnaseer
  • asnaseer
good, now lets first simplify this by adding 1 to both sides to get rid of the -1 from the left-hand-side
asnaseer
  • asnaseer
\[(x+1)^2-1+y^2+4y=20\]\[\therefore (x+1)^2-1+y^2+4y+1=20+1\]\[(x+1)^2+y^2+4y=21\]following so far?
Godlovesme
  • Godlovesme
yes :)
asnaseer
  • asnaseer
great! so now we have:\[(x+1)^2+y^2+4y=21\]lets now concentrate on the terms involving \(y\), we have:\[y^2+4y\]here again we know that:\[(y+b)^2=y^2+2by+b^2\]so can you think of what value we need for \(b\) so that the \(2by\) term becomes \(4y\)?
Godlovesme
  • Godlovesme
2 :3
asnaseer
  • asnaseer
perfect! :)
asnaseer
  • asnaseer
so we now have:\[(y+2)^2=y^2+4y+4\]but we are after just \(y^2+4y\), so we need to subtract 4 from both sides, i.e.:\[(y+2)^2=y^2+4y+4\]\[\therefore (y+2)^2-4=y^2+4y+4-4=y^2+4y\]agreed?
Godlovesme
  • Godlovesme
yes :D
asnaseer
  • asnaseer
ok, now recall we originally were given:\[x^2 + 2x + y^2 + 4y = 20\]which we transformed using \((x+1)^2-1=x^2+2x\) into:\[(x+1)^2+y^2+4y=21\]and now we also know that:\[(y+2)^2-4=y^2+4y\]so if we now replace \(y^2+4y\) with the new expression we just found for it, then we get:\[(x+1)^2+(y+2)^2-4=21\]agreed?
Godlovesme
  • Godlovesme
agreed c:
asnaseer
  • asnaseer
now again we can simplify this bu adding 4 to both sides as follows:\[(x+1)^2+(y+2)^2-4=21\]\[\therefore (x+1)^2+(y+2)^2-4+4=21+4\]\[\therefore (x+1)^2+(y+2)^2=25\]
asnaseer
  • asnaseer
agreed?
Godlovesme
  • Godlovesme
i get it now!! so the center will be (-1,-2) and the radius will be 5? :D
asnaseer
  • asnaseer
perfect! I was going to add one more step saying you have to be careful now as we are trying to get to this form:\[(x-h)^2+(y-k)^2=r^2\]so we must rewrite our equation as:\[(x-(-1))^2+(y-(-2))^2=5^2\]but I guess you beat me to it! :D
Godlovesme
  • Godlovesme
OMG THANK YOU SO MUCHHH!!!! you're a life saver, i really appreciate your help :))) <3
asnaseer
  • asnaseer
yw my friend. I really appreciate people who are genuinely interested in learning - keep up the good work! :)
Godlovesme
  • Godlovesme
ahh you're awesome!! I spend more than an hour trying to figure this out lol u solved it in less that half an hour XD
asnaseer
  • asnaseer
ok, bye for now - take care my friend :)
Godlovesme
  • Godlovesme
thx and you too :)

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