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Math Help - Equal roots

  1. #1
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    Equal roots

    Hi! I need some help on understanding this proof:

    If the equation  x^2 + 2(k+2)x +9k = 0 has equal roots, find k.

    The condition for equal roots gives
     (k+2)^2 = 9k
     k^2 - 5k + 4 =0
     (k-4)(k-1) = 0
     k=4 , k=1

    The proof makes sense: When k = 4, for example, we get  x^2 + 12x + 36 and this will give equal roots if  b^2 - 4ac = 0 which it does since  144 - 144 = 0

    But I still don't understand where the first line of the proof,
     (k+2)^2 = 9k , comes from

    From Higher Algebra, H.S Hall & Knight
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  2. #2
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    Do you remember "Completing the Square"? It is just a step from that process, usually verbalized, "The square of one-half the coefficient of the linear term". Probably, it is why the '2' is part of the linear term, so that finding half of it will be more convenient.
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  3. #3
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    Here: a = 1, b = 2(k+2), c = 9k.

    Plug this into the discriminant:
    b^{2} - 4ac = 0
    \big[2(k+2)\big]^2 - 4(1)(9k) = 0
    4(k+2)^{2} = 36k (squared the first term recalling that (ab)^{n} = a^nb^n, moved the 2nd term to the other side)
    (k+2)^2 = 9k
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    Quote Originally Posted by storchfire1X View Post
    The proof makes sense: When k = 4, for example, we get  x^2 + 12x + 36 and this will give equal roots if  b^2 - 4ac = 0 which it does since  144 - 144 = 0
    You said it yourself: the equation will have equal roots iff the discriminant b^2 - 4ac = 0.

    So, we have

    b^2 - 4ac = 0

    \Rightarrow\left[2\left(k + 2\right)\right]^2 - 4(1)(9k) = 0

    \Rightarrow4\left(k + 2\right)^2 = 4(9k)

    \Rightarrow\left(k + 2\right)^2 = 9k

    Edit: Beaten by 21 minutes! Ouch.
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