The quadratic equation x^2-6x+14=0 has roots alpha and beta. a) Write down the value of alpha+beta and the value of alpha*beta. b) Find a quadratic equation, with integer coefficients which has roots alpha/beta and beta/alpha.

a) We know that, for a general quadratic equation that can be written as : ax²+bx+c=0, the sum of its roots (S) is equal to -b/a and the product of its roots (P) is equal to c/a (which can easily be verified). We have S=-b/a and P=c/a

In this example, a=1, b=-6 and c=14. Therefore we have : S=alpha+beta=6/1=6 and P=alpha*beta=14/1=14

b) The equation we want to find is a quadratic equation, that can be written as : ax²+bx+c=0

We know that :

S=alpha/beta+beta/alpha=(alpha²+beta²)/(alphabeta)=-b/a and P=alpha/betabeta/alpha=1=c/a

<=> [(alpha+beta)²-2alphabeta]/(alpha*beta)=-b/a and c=a

<=> (6²-2*14)/14=8/14=4/7=-b/a and c=a (using the results we found in question a)

<=> b=-(4/7)*a and c=a

We know that ax²+bx+c=0. So using the results we just found, we have :

ax²-(4/7)a+a=0 <=> a(x²-(4/7)*x+1)=0 <=> x²-4/7x+1=0 <=> 7x²-4x+7=0 (because we want integer coefficients).