if alpha (α) and beta (β) are the roots of 2x^2-7x+10=0, find: (α) + (β) (α) x (β) (α)^2 + (β)^2 (α)^2 x (β) + (β)^2 x (α)

We'll find the roots of the quadratic using the
formula:

4ac))/(2a)

4ac))/(2a)

a,b,c are the coefficients of the
quadratic:

a = 2 ; b = -7 and c =
10

80))/(4)

We notice that the result of the difference under
the radical sign is negative, therefore the equation has complex
roots.

and
isqrt31)/4

We'll put and
beta

isqrt31)/4

We'll eliminate imaginary
parts:

14/4

7/2

We'll calculate the product of roots, that is a
special product which returns a difference of two
squares:

31i^2)/16

But =>
(49+31)/16alpha*beta = 5

We'll calculate
:

2alpha*beta

=>

We'll calculate
beta^2*alpha = alpha*beta(alpha + beta)

beta^2*alpha = 5*7/2 = 35/2

Therefore, the
requested results are:
alpha^2*beta + beta^2*alpha = 35/2