1. ## 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).

2. Originally Posted by cakeboby
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).

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)