Let |sgt and |tgt denote orthonormal states. Let |Psi_1> =|sgt+2i|tgt and |Psi_2gt =2|sgt+x|tgt . What must the value of x be so that |Psi_1gt and |Psi_2gt are orthogonal?

The states |psi_1> and |psi_2> will be orthogonal if and only if the product = 0 .Therefore, to find the value of x that makes the states orthogonal, we will need to substitute the definitions of the functions into the equation
= 0
and solve the resulting equation for x.
We have
|psi_1> = |s> + 2i |t> and
|psi_2> = 2|s> + x|t>
The complex conjugate is So = (1)(2) + (-2i)(x) = 0
Because |s> and |t> are orthonormal states, we know that by definition
= = 1 so
2 - 2ix = 0
2ix = 2 and
x = 1/i .
We have to be careful with - signs.
Since i = sqrt(-1)
We know i^2 = -1

and so -i^2 = 1 .
Thus x = -i^2/i
or x =-i , and
|psi_2> = 2 |s> - i |t> .
Substituting this value into the equation above, we find
= 2 - 2 = 0 so the value of -i does cause the states to be orthogonal.