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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.
 one year ago
 one year 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.
 one year ago
 one year ago

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AccessDeniedBest ResponseYou've already chosen the best response.1
I 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 \(c^2  4d\): \( \Large a + bi = \frac{\neg c + i \sqrt{4d  c^2}}{2} \) \( \Large a + bi = \neg \frac{c}{2} + \frac{\sqrt{4d  c^2}}{2} i \)
 one year ago

AccessDeniedBest ResponseYou've already chosen the best response.1
So, 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. :)
 one year ago

sabika13Best ResponseYou've already chosen the best response.0
okay that makes sense, but can you explain the "i", i dont understand why  1 gets the imaginary part??
 one year ago

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

sabika13Best ResponseYou've already chosen the best response.0
does "i" always equal 1 ?
 one year ago

AccessDeniedBest ResponseYou've already chosen the best response.1
\(i =\sqrt{1}\), not just 1.
 one year ago

sabika13Best ResponseYou've already chosen the best response.0
ohh i get it! thanks!
 one year ago

AccessDeniedBest ResponseYou've already chosen the best response.1
You're welcome! :) Were you able to figure out the rest?
 one year ago

sabika13Best ResponseYou've already chosen the best response.0
im going to try it now
 one year ago

sabika13Best ResponseYou've already chosen the best response.0
for quadratic formula, isnt its x= b +/ square roots... why did you just write + ?
 one year ago

AccessDeniedBest ResponseYou've already chosen the best response.1
We'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?
 one year ago

sabika13Best ResponseYou've already chosen the best response.0
i got a^2 + b^2 = d, but i cant get 2a+c =0 2(c/2) +\[\sqrt{4dc ^{2}}\] =0 c + dc = 0?
 one year ago

AccessDeniedBest ResponseYou've already chosen the best response.1
hm... 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
 one year ago

sabika13Best ResponseYou've already chosen the best response.0
yeaah! I didnt see that, I got it thanks so much!
 one year ago

AccessDeniedBest ResponseYou've already chosen the best response.1
You're welcome! :)
 one year ago
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