Thread: 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.

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).

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