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

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})