N 155
we have
x^2+px+q=0
r and s are roots
that means
x^2+px+q=(x-r)(x-s)
(x-r)(x-s)=x^2-xs-xr+sr=x^2-(s+r)x+sr
so
x^2+px+q=x^2-(s+r)x+sr
that means
p=-(s+r) -----> equation 1
q=sr -----> equation 2
Part 1
r^2+s^2
squared equation 1
p^2=-(s+r)^2
p^2=(s^2+2sr+r^2)
Remember that
q=sr (equation 2)
so
p^2=(s^2+2q+r^2)
p^2=s^2+r^2+2q
r^2+s^2=p^2-2q
Part 4
r^4+s^4
we have
(r^2+s^2)=(-p^2-2q)
squared both sides
(r^2+s^2)^2=(-p^2-2q)^2
r^4+2r^2*s^2+s^4=p^4+4p^2q+4q^2
r^4+s^4=(p^4+4p^2q+4q^2)-2r^2*s^2
Remember that
q=sr
so
2r^2*s^2=2q^2
substitute
r^4+s^4=(p^4+4p^2q+4q^2)-2q^2
r^4+s^4=p^4+4p^2q+2q^2
Part 3
r^2s+rs^2
p=-(s+r) -----> equation 1
q=sr -----> equation 2
Multiply p*q
p*q=-(s+r)*(sr)
p*q=-(s^2r+r^2s)
therefore
r^2s+rs^2=-p*q
Part 2
√(p^2 - 4q)
we have
p^2=(s+r)^2
p^2=(s^2+2q+r^2)
4q=4sr
(s^2+2q+r^2)-4sr
2q=2sr
so
(s^2+2sr+r^2)-4sr
s^2-2sr+r^2
rewrite as perfect squares
√(r-s)^2=r-s