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anonymous
 one year ago
**please help!**
Using the following equation, find the center and radius of the circle.
x^2  4x + y^2 + 8y = 4
anonymous
 one year ago
**please help!** Using the following equation, find the center and radius of the circle. x^2  4x + y^2 + 8y = 4

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anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I would complete the square twice, once for a quadratic in x and once for a quadratic in y.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0can you show me step by step?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0In other words, Add 4 to the 4^24x to create a perfect square trinomial. Then add 16 to the y^2 + 8y to make another perfect square trinomial. Then, because 20 is added to the left hand side, 20 must also be added to th right hand side.\[x^24x+y^2+8y=4\]\[x^24x+4+y^2+8y+16=4+20\]I combined some stuff. Do you understand what I've done?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Now, consider the LHS as two trinomials. We can group them to make it easier to follow. Plus, we can simplify the RHS\[\left( x^24x+4 \right)+\left( y^2+8y+16 \right)=16\]OK with that?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0OK. Now, the two trinomials on the LHS are perfect squares. Can you factor them and write them as squares?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0would it be (4^2−4x+4)+(20^2+8y+16)=16

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Not quite. Let's slow down a bit and take it step by step. The first trinomial on the LHS is\[x^24x+4\]Can you factor this into the product of two binomials?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0x^24x+44+20? im really bad at math and I'm home schooled so it's even tougher

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Have you multiplied binomials together before? Stuff like\[\left( x+2 \right)\left( x3 \right) = x^2x6\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Oh, then we have a problem. Factoring is the reverse of the above. Let me think of there's another way to get at this...

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0So you haven't used this technique of 'completing the square' before either?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0nope. I took basics of alegebra then was thrown into geometry

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0OK. Do you know what the equation of a circle looks like?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0That's very good, but the radius must also squared. For a circle with center at the origin (0, 0), it's equation would be\[x^2+y^2=r^2\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0IN your question, however, the center of the circle is NOT at the origin. So, for a circle with its center at some point (a, b), its equation looks like\[\left( xa \right)^2 + \left( yb \right)^2 = r^2\]Notice that, if the center is at the origin, a=b=0 and we get the equation you proposed earlier. Understand?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Alright. So, I've been tying to help you get the equation you were given in the problem\[x^24x+y^2+8y=4\]to look like\[\left( xa \right)^2+\left( yb \right)^2 = r^2\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0So far, we've done a couple of things and are at the stage where we have\[\left( x^24x+4 \right)+\left( y^2+8y+16 \right)=16\]Look at the RHS. What number squared is equal to 16?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Good. So that's the radius. OK with that much?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Good. Now I'm going to cheat a bit and factor those trinomials for you. We have\[\left( x^24x+x \right)+\left( y^2+8x+16 \right) = 4^2\]\[\left( x2 \right)^2+\left( y+4 \right)^2=4^2\]Compare that with\[\left( xa \right)^2+\left( yb \right)^2=r^2\]Can you tell what a and b are?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0yes a is 2 and b is 4

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0You're right about a, but look carefully at the signs and try b again.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Nope. If b=4, then you'd have (y  4)^2. But you want (y + 4)^2. What does b have to be?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0That's right.\[\left( y\left( 4 \right) \right)^2 = \left( y+4 \right)^2\]So b must be 4.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0So now you know a, b, and r. Where's the center of the circle?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0No. We know that a=2, b=4 and r=4. The center of the circle is at coordinates (a, b) and the radius is r. So where;s the center of the circle?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Correct. And what's the radius?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0There's your answer.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0thanks soooooo muchhh!!!

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0You're welcome. I would encourage you to get some help learning to multiply and factor polynomials. Those skills will be necessary to move forward. Good luck!
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