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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.

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  1. AccessDenied
    • 2 years ago
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    I believe that we could use quadratic formula here, and equate the like-parts (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 \(c^2 - 4d\): \( \Large a + bi = \frac{\neg c + i \sqrt{4d - c^2}}{2} \) \( \Large a + bi = \neg \frac{c}{2} + \frac{\sqrt{4d - c^2}}{2} i \)

  2. AccessDenied
    • 2 years ago
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    So, if we equate the like-parts 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. :)

  3. sabika13
    • 2 years ago
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    okay that makes sense, but can you explain the "i", i dont understand why - 1 gets the imaginary part??

  4. AccessDenied
    • 2 years ago
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    we factor out the -1 from under a square root, so the square root of the -1 = i \( \sqrt{c^2 - 4d} = \sqrt{(-1)(4d - c^2)} = \sqrt{-1}\sqrt{4d - c^2}\)

  5. sabika13
    • 2 years ago
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    does "i" always equal -1 ?

  6. AccessDenied
    • 2 years ago
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    \(i =\sqrt{-1}\), not just -1.

  7. sabika13
    • 2 years ago
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    ohh i get it! thanks!

  8. AccessDenied
    • 2 years ago
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    You're welcome! :) Were you able to figure out the rest?

  9. sabika13
    • 2 years ago
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    im going to try it now

  10. sabika13
    • 2 years ago
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    for quadratic formula, isnt its x= -b +/- square roots... why did you just write + ?

  11. AccessDenied
    • 2 years ago
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    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?

  12. sabika13
    • 2 years ago
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    yes!!

  13. sabika13
    • 2 years ago
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    i got a^2 + b^2 = d, but i cant get 2a+c =0 2(-c/2) +\[\sqrt{4d-c ^{2}}\] =0 -c + d-c = 0?

  14. AccessDenied
    • 2 years ago
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    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

  15. sabika13
    • 2 years ago
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    yeaah! I didnt see that, I got it thanks so much!

  16. AccessDenied
    • 2 years ago
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    You're welcome! :)

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