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Thread: two quadratic equation problems

  1. #1
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    two quadratic equation problems

    1. Prove
    $\displaystyle a^2 + b^2+c^2-ab-ac-bc > 0$

    2. Let $\displaystyle \alpha $ and $\displaystyle \beta $ be the roots of the equation $\displaystyle x^2 +kx +(p-k-1)=0 $, where p and k are positive real numbers.
    (i) Find the quadratic equation whose roots are $\displaystyle \alpha -1 $and $\displaystyle \beta -1 $
    (ii) If p is prime and $\displaystyle \alpha $, $\displaystyle \beta$ are integers, show that k = p - 1

    For question 2 I have trouble in (ii).

    Thanks in advance.
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  2. #2
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    Quote Originally Posted by cakeboby View Post
    2. Let $\displaystyle \alpha $ and $\displaystyle \beta $ be the roots of the equation $\displaystyle x^2 +kx +(p-k-1)=0 $, where p and k are positive real numbers.
    (i) Find the quadratic equation whose roots are $\displaystyle \alpha -1 $and $\displaystyle \beta -1 $
    (ii) If p is prime and $\displaystyle \alpha $, $\displaystyle \beta$ are integers, show that k = p - 1

    For question 2 I have trouble in (ii).

    Thanks in advance.
    Hi

    You should find in (i) $\displaystyle x^2 + (k+2)x + p = 0$

    If p is prime then $\displaystyle \alpha-1=1$ and $\displaystyle \beta-1=p$ (or the other way)

    I let you finish
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