Find the Values for k

Find the values of k such that the equation has exactly one real root.

K(x^2) + (sqrt{2k})*x + 6 = 0

Set the expression for the discriminant equal to zero and solve for k.
(I wasn't quite sure whether the capital K was supposed to be different from the lower case k. If it was, then that is the assumption I made when working it out below. If it wasn't, then there is no solution since solving gives k = 0, which just leaves the equation 6 = 0, which is not only not quadratic but also never true)

I have this part in one of your other questions but have it here also for reference:
The discriminant is given by:
√(b^2 - 4*a*c)

For quadratic equations in the form:
ax^2 + bx + c = 0

So in this problem:
a = K
b = √(2k)
c = 6


The discriminant:
√( √(2k)^2 - 4 * K * 6 )
√( 2k - 24K )

Set it equal to zero:
√( 2k - 24K ) = 0
2k - 24K = 0
2k = 24K

k = 12K <------------------------------Answer

if there is only one root then discriminant is 0
ie (sqrt{2k})^2 -4*k*6 is 0 ie 2k-24k is 0 ie k is 0

If you set k to zero, the equation won't be valid since 6 ≠ 0.