SQUARE ROOTS OF NON-COMPLEX NUMBERS

• Let us start with the complex number where k is positive

• c= a + b√k (1)
  

• Let us assume that a square root of c is p + q√k where p and q are real

• (p + q√k)² = a + b√k (2)
  

• Equating the real and radical parts gives us the two equations

• p² + kq² = a (3)
  
• 2pq = b (4)
  

• We must have p ≠ 0 since b ≠ 0. Solving equation (4) for q gives

• q =  b  (5)
        2p

• and we can substitute this value for q into equation (4) to get

• p² + k( b²) = a (6)
   (4p²)

• or

• 4p⁴ − 4ap² + kb² = 0 (7)
  

• This is a quadratic in p², so we can solve for p² using the quadratic formula.

• p² =  4a ±√16a2 − 16kb2  (8)
    8

• and after factoring out

• p² =  a ±√a2 − kb2  (9)
    2

• so that

• p =  √a ±√a2 − kb2  (10)
     √ 2

• At this point r which contains k is set to the following:

• r = ±√a2 − kb2  (11)
  

• Giving p and consequently x:

• p = x =  √a ± r   (12)
          √ 2

• The value of b from (11) is

• b =   ±√a2 − r2  (13)
       √ k

• Substituting p from equation (12) into equation (5)

• q =       b           (14)
        2√(a ± r)/2  
  

• Multiplying the top and bottom of equation (14) by √a ∓ r   and substituting the value for b from (13) gives the following:

• q =  √a ∓ r     (15)
     √ 2k

• This means that whenever the radical in equation (12) is √a + r   the value of the radical in (15) would be √a − r   and vice versa. Moreover, substituting x in (14) gives (16).

• y =      b     (16)
       2x

Example I:

4036 + 2322√3

Find r

r = √40362 − 3×23222  = 338

Find x then y  (note √(a + r  )/2 does not give an integer value).

x = √4036 − 338     = 43
√ 2

y =     2322    = 27
2 × 43

Giving the answer

(43 + 27√3)²

---

Example II:

229 − 132√3

Find r

r = √2292 − 3×1322  = 13

Find x then y  (note √(a − r  )/2 does not give an integer value).

x = √229 + 13     = 11
√ 2

y =     −132    = −6
2 × 11

Giving the answer

(11 − 6√3) ²

---

Example III:

15675 + 11050√2

Find r

r = √156752 − 2×110502  = 1225

Find x then y  (note √(a + r  )/2 does not give an integer value).

x = √15675 − 1225     = 85
√ 2

y =     11050    = 65
2 × 85

Giving the answer

(85 + 65√2) ²