If a and b are the root of 10x^2 - 13x +3 =0 find a+b (2) ab (3) a^2+b^2 (4)a^3+b^3 (5) a-b

a and b are the roots of 10x^2 - 13x + 3 = 0. By using the sum and product of roots formulae, we'll get

a + b = -(-13)/10 = 13/10
ab = 3/10

This itself solves (1) and (2).

(3) a^2 + b^2
= (a + b)^2 - 2ab
= (13/10)^2 - 2(3)/10
= 169/100 - 6/10
= 109/100

(4) a^3 + b^3
= (a + b)(a^2 - ab + b^2)
= (a + b) ((a + b)^2 - 3ab)
= 13/10 ((13/10)^2 - 3(3/10))
= 13/10 (169/100 - 9/10)
= 13/10 (79/100)
= 1027/1000

(5) a - b
= √(a - b)^2
= √(a^2 + b^2 - 2ab)
= √[(a + b)^2 - 4ab]
= √[(13/10)^2 - 4(3/10)]
= √(169/100 - 12/10)
= √49/100
= 7/10