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sabika13
 2 years ago
If a + bi is a root of the quadratic equation x^2 + cx + d = 0, then show that a^2 + b^2 = d and 2a+c=0.
sabika13
 2 years ago
If a + bi is a root of the quadratic equation x^2 + cx + d = 0, then show that a^2 + b^2 = d and 2a+c=0.

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AccessDenied
 2 years ago
Best ResponseYou've already chosen the best response.1I believe that we could use quadratic formula here, and equate the likeparts (real and imaginary): \( \Large { x = \frac{\neg b + \sqrt{b^2  4ac}}{2a} } \) We'll only need the positive part because x = a + bi, it is adding the bi. So, this equation would look like this: \( \Large a + bi = \frac{\neg c + \sqrt{c^2  4d}}{2} \) To get the imaginary part, we need to factor out a 1 from \(c^2  4d\): \( \Large a + bi = \frac{\neg c + i \sqrt{4d  c^2}}{2} \) \( \Large a + bi = \neg \frac{c}{2} + \frac{\sqrt{4d  c^2}}{2} i \)

AccessDenied
 2 years ago
Best ResponseYou've already chosen the best response.1So, if we equate the likeparts here, we get: a = c/2 b = sqrt(4d  c^2)/2 (The i's can divide off) From there, it takes some clever algebra work to get the two results you wanted to show. :)

sabika13
 2 years ago
Best ResponseYou've already chosen the best response.0okay that makes sense, but can you explain the "i", i dont understand why  1 gets the imaginary part??

AccessDenied
 2 years ago
Best ResponseYou've already chosen the best response.1we factor out the 1 from under a square root, so the square root of the 1 = i \( \sqrt{c^2  4d} = \sqrt{(1)(4d  c^2)} = \sqrt{1}\sqrt{4d  c^2}\)

sabika13
 2 years ago
Best ResponseYou've already chosen the best response.0does "i" always equal 1 ?

AccessDenied
 2 years ago
Best ResponseYou've already chosen the best response.1\(i =\sqrt{1}\), not just 1.

AccessDenied
 2 years ago
Best ResponseYou've already chosen the best response.1You're welcome! :) Were you able to figure out the rest?

sabika13
 2 years ago
Best ResponseYou've already chosen the best response.0im going to try it now

sabika13
 2 years ago
Best ResponseYou've already chosen the best response.0for quadratic formula, isnt its x= b +/ square roots... why did you just write + ?

AccessDenied
 2 years ago
Best ResponseYou've already chosen the best response.1We're only dealing with one root when we use 'x = a + bi,' the positive square root here. The other root is 'x = a  bi', which is where the negative square root is used. Does that make sense?

sabika13
 2 years ago
Best ResponseYou've already chosen the best response.0i got a^2 + b^2 = d, but i cant get 2a+c =0 2(c/2) +\[\sqrt{4dc ^{2}}\] =0 c + dc = 0?

AccessDenied
 2 years ago
Best ResponseYou've already chosen the best response.1hm... a = c/2 Here we should have enough to get 2a + c = 0, we have all our variables in an equation! 2a = c we can add c to both sides 2a + c = 0

sabika13
 2 years ago
Best ResponseYou've already chosen the best response.0yeaah! I didnt see that, I got it thanks so much!

AccessDenied
 2 years ago
Best ResponseYou've already chosen the best response.1You're welcome! :)
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