F (x) = - 3x + 5x2 + 8 has roots.

it has no root

first make it common quadratic form

y = 5x² - 3x + 8

we take

a = 5

b = - 3

c = 8

the to prove it doesn't have root. we using discriminant

D = b² - 4ac

D = (-3) ² - 4*5*8

D = 9 - 160

D = - 151

D > 0 means the equation has 2 roots

D = 0 means the equation has 1 root

D < 0 do not have root because it doesn't have intercept on x-axis