Find the square root of 170 – 20(sqrt{30})

A.
2 (sqrt{10}) – 5(sqrt{3}) 
B.
2 (sqrt{5}) – 5(sqrt{6}) 
C.
5 (sqrt{10}) – 2(sqrt{3}) 
D.
3 (sqrt{5}) – 8(sqrt{6})
Correct Answer: Option B
Explanation
(sqrt{170 – 20 sqrt{30}} = sqrt{a} – sqrt{b})
Squaring both sides,
(170 – 20sqrt{30} = a + b – 2sqrt{ab})
Equating the rational and irrational parts, we have
(a + b = 170 … (1))
(2 sqrt{ab} = 20 sqrt{30})
(2 sqrt{ab} = 2 sqrt{30 times 100} = 2 sqrt{3000} )
(ab = 3000 … (2))
From (2), (b = frac{3000}{a})
(a + frac{3000}{a} = 170 implies a^{2} + 3000 = 170a)
(a^{2} – 170a + 3000 = 0)
(a^{2} – 20a – 150a + 3000 = 0)
(a(a – 20) – 150(a – 20) = 0)
(text{a = 20 or a = 150})
(therefore b = frac{3000}{20} = 150 ; b = frac{3000}{150} = 20)
(sqrt{170 – 20sqrt{30}} = sqrt{20} – sqrt{150}) or (sqrt{150} – sqrt{20})
= (2sqrt{5} – 5sqrt{6}) or (5sqrt{6} – 2sqrt{5})