SOLUTION: If (√3 - 1) is a root of the equation 2x^2-2kx+4 = 0, then k equals... A)2√3-1 B)√3-2 C)2√3 D)√3 E)√3-1

Algebra ->  Equations -> SOLUTION: If (√3 - 1) is a root of the equation 2x^2-2kx+4 = 0, then k equals... A)2√3-1 B)√3-2 C)2√3 D)√3 E)√3-1      Log On


   

Question 1128329: If (√3 - 1) is a root of the equation 2x^2-2kx+4 = 0, then k equals...
A)2√3-1 B)√3-2 C)2√3 D)√3 E)√3-1
Answer by ikleyn(52786) About Me  (Show Source):
You can put this solution on YOUR website!
.
The equation

    2x^2 - 2kx + 4 = 0 

is equivalent to  (after dividing both sides by 2)


    x^2 - kx + 2 = 0.      (1)


We are given that  sqrt%283%29-1  is the root of the original equation; hence, the equation (1) with the leading coefficient 1 has this root, too.


Then, applying the Vieta's theorem, the other root of the equation (1) is


    2%2F%28sqrt%283%29-1%29 = 2%2F%28sqrt%283%29-1%29.%28sqrt%283%29%2B1%29%2F%28sqrt%283%29%2B1%29 = %282%2Asqrt%283%29%2B1%29%2F%28%28sqrt%283%29%29%5E2-1%5E2%29 = %282%2A%28sqrt%283%29%2B1%29%29%2F2 = sqrt%283%29%2B1.


Thus we know BOTH ROOTS of the equation (1) (even without solving it explicitly (!) ). They are


    sqrt%283%29-1  and  sqrt%283%29%2B1.


Again, according to Vieta's theorem, the sum of these roots is the coefficient at x in equation (1) taken with the opposite sign:


    k = sqrt%283%29-1 + sqrt%283%29%2B1 = 2%2Asqrt%283%29.


Answer.  k = 2%2Asqrt%283%29.

Solved.   //   Option C).