Question

Find a and b
(root2.3 - root0.69)/(root2.3 + root0.69)
= a + b root30

Answer 1

multiply by (√2.3 - √.69) top and bottom and you get
(√2.3 - √.69)^2 / ((√2.3 + √.69)(√2.3 - √.69))
= 2.3 - 2√1.587 + .69) / (2.3-.69)
= (2.99 - 2√1.587)/1.61
Now how you expect to get √30 out of that is beyond me.
Maybe you can figure it out

Answer 2

perhaps we have to match rationals with rationals and irrationals with irrationals?
let's give it a try, picking up at oobleck's last step

(2.99 - 2√1.587)/1.61 = a + b√30

so a = 2.99/1.61 = 13/7
b√30 = -2√.1.587 / 1.61
b = -2√.1.587 / (1.61√30) = appr -.2857

a = 13/7
b = -.2857

so a + b√30 = 13/7 - .2857√30

check:
13/7 - .2857√30 = appr .29222..
and
(2.99 - 2√1.587)/1.61 = 2.9222..
ok then!